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Question 1 of 57
Alan and Ben are working on a science project. Together, they can finish it in 35 days. If Alan works alone for 40 days, Ben will take a further 28 days to complete the project. If Alan works alone for 30 days, how many more days will Ben take to complete the project?
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\begin{align*} \text{40 days by }A + 28\text{ days by }B &= 1 \text{ project} \\ \text{35 days by }A + 35\text{ days by }B &= 1 \text{ project} \end{align*} Solve above 2 equations get ratio $A:B=7:5$. $7+5=12$. \begin{equation} B : 35 \times 12 /5 = 84 \text{ days alone,} \end{equation} so in one day $B$ does $1/84$ work. \begin{equation} A: 35 \times 12 /7 = 60 \text{ days alone,} \end{equation} so in one day $A$ does $1/60$ work $($or $1/35-1/84=1/60)$. \begin{equation} A: 1/60 \times 30\text{ days} = \frac{1}{2} \text{ of work.} \end{equation} So $\frac{1}{2}$ of work left for $B$. \begin{equation} B: 1/84 \times \text{ number of days } = \frac{1}{2} \end{equation} so number of days $=\frac{1}{2}\times 84 = 42$.
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\begin{align*} \text{40 days by }A + 28\text{ days by }B &= 1 \text{ project} \\ \text{35 days by }A + 35\text{ days by }B &= 1 \text{ project} \end{align*} Solve above 2 equations get ratio $A:B=7:5$. $7+5=12$. \begin{equation} B : 35 \times 12 /5 = 84 \text{ days alone,} \end{equation} so in one day $B$ does $1/84$ work. \begin{equation} A: 35 \times 12 /7 = 60 \text{ days alone,} \end{equation} so in one day $A$ does $1/60$ work $($or $1/35-1/84=1/60)$. \begin{equation} A: 1/60 \times 30\text{ days} = \frac{1}{2} \text{ of work.} \end{equation} So $\frac{1}{2}$ of work left for $B$. \begin{equation} B: 1/84 \times \text{ number of days } = \frac{1}{2} \end{equation} so number of days $=\frac{1}{2}\times 84 = 42$.
A mall escalator moves upwards at a constant rate from the $1^{\text{st}}$ to $3^{\text{rd}}$ floor. Chiranjit walks up the escalator at a constant rate of 2 steps per second, and arrives at the third floor after 40 seconds. Debdeep walks up the escalator at a constant rate of 3 steps every 2 seconds, and arrives at the third floor after 50 seconds. How many steps are there on the escalator?
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Let the speed of escalator be $e$ \begin{align*} & 40 \times 2 + 40 e = 50\times \frac{3}{2}+50e \\ & e=0.5 \text{ steps}/s \\ & 40\times 2 + 40e = 100 \text{ steps} \\ & \text{or } 50\times \frac{3}{2} + 50e = 100 \text{ steps} \end{align*}
Find the value of \begin{equation*} 1^3 + 2^3 + 3^3+\cdots+13^3 + 14^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 14^3 \\ & = (1+2+3+\cdots+14)^2 = 11025 \end{align*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 14^3 \\ & = (1+2+3+\cdots+14)^2 = 11025 \end{align*}
Find the value of \begin{equation*} 1^3 + 2^3 + 3^3+\cdots+14^3 + 15^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 15^3 \\ & = (1+2+3+\cdots+15)^2 = 14400 \end{align*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 15^3 \\ & = (1+2+3+\cdots+15)^2 = 14400 \end{align*}
Find the value of \begin{equation*} 1^3 + 2^3 + 3^3+\cdots+15^3 + 16^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 16^3 \\ & = (1+2+3+\cdots+16)^2 = 18496 \end{align*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 16^3 \\ & = (1+2+3+\cdots+16)^2 = 18496 \end{align*}
Find the value of \begin{equation*} 1^3 + 2^3 + 3^3+\cdots+16^3 + 17^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 17^3 \\ & = (1+2+3+\cdots+17)^2 = 23409 \end{align*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 17^3 \\ & = (1+2+3+\cdots+17)^2 = 23409 \end{align*}
Find the value of \begin{equation*} 1^3 + 2^3 + 3^3+\cdots+17^3 + 18^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 18^3 \\ & = (1+2+3+\cdots+18)^2 = 29241 \end{align*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 18^3 \\ & = (1+2+3+\cdots+18)^2 = 29241 \end{align*}
Find the value of \begin{equation*} 1^3 + 2^3 + 3^3+\cdots+18^3 + 19^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 19^3 \\ & = (1+2+3+\cdots+19)^2 = 36100 \end{align*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 19^3 \\ & = (1+2+3+\cdots+19)^2 = 36100 \end{align*}
Find the value of \begin{equation*} 1^3 + 2^3 + 3^3+\cdots+19^3 + 20^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 20^3 \\ & = (1+2+3+\cdots+20)^2 = 44100 \end{align*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 20^3 \\ & = (1+2+3+\cdots+20)^2 = 44100 \end{align*}
Find the value of \begin{equation*} 1^3 + 2^3 + 3^3+\cdots+20^3 + 21^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 21^3 \\ & = (1+2+3+\cdots+21)^2 = 53361 \end{align*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 21^3 \\ & = (1+2+3+\cdots+21)^2 = 53361 \end{align*}
Find the value of \begin{equation*} 1^3 + 2^3 + 3^3+\cdots+21^3 + 22^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 22^3 \\ & = (1+2+3+\cdots+22)^2 = 64009 \end{align*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 22^3 \\ & = (1+2+3+\cdots+22)^2 = 64009 \end{align*}
Find the value of \begin{equation*} 1^3 + 2^3 + 3^3+\cdots+22^3 + 23^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 23^3 \\ & = (1+2+3+\cdots+23)^2 = 76176 \end{align*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 23^3 \\ & = (1+2+3+\cdots+23)^2 = 76176 \end{align*}
Find the value of \begin{equation*} 1^3 + 2^3 + 3^3+\cdots+23^3 + 24^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 24^3 \\ & = (1+2+3+\cdots+24)^2 = 90000 \end{align*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 24^3 \\ & = (1+2+3+\cdots+24)^2 = 90000 \end{align*}
Find the value of \begin{equation*} 2^3 + 3^3 + 4^3+\cdots+13^3 + 14^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 14^3 \\ & = (1+2+3+\cdots+14)^2 = 11025 \end{align*} \begin{align*} \therefore\quad 2^3 + 3^3 + 4^3+ \cdots+ 13^3 + 14^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 13^3 + 14^3\right) - \left( 1^3 \right) \\ &=11025 - \left( 1 \right)^2 =11024 \end{align*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 14^3 \\ & = (1+2+3+\cdots+14)^2 = 11025 \end{align*} \begin{align*} \therefore\quad 2^3 + 3^3 + 4^3+ \cdots+ 13^3 + 14^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 13^3 + 14^3\right) - \left( 1^3 \right) \\ &=11025 - \left( 1 \right)^2 =11024 \end{align*}
Find the value of \begin{equation*} 2^3 + 3^3 + 4^3+\cdots+14^3 + 15^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 15^3 \\ & = (1+2+3+\cdots+15)^2 = 14400 \end{align*} \begin{align*} \therefore\quad 2^3 + 3^3 + 4^3+ \cdots+ 14^3 + 15^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 14^3 + 15^3\right) - \left( 1^3 \right) \\ &=14400 - \left( 1 \right)^2 =14399 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 15^3 \\ & = (1+2+3+\cdots+15)^2 = 14400 \end{align*} \begin{align*} \therefore\quad 2^3 + 3^3 + 4^3+ \cdots+ 14^3 + 15^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 14^3 + 15^3\right) - \left( 1^3 \right) \\ &=14400 - \left( 1 \right)^2 =14399 \end{align*}
Find the value of \begin{equation*} 2^3 + 3^3 + 4^3+\cdots+15^3 + 16^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 16^3 \\ & = (1+2+3+\cdots+16)^2 = 18496 \end{align*} \begin{align*} \therefore\quad 2^3 + 3^3 + 4^3+ \cdots+ 15^3 + 16^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 15^3 + 16^3\right) - \left( 1^3 \right) \\ &=18496 - \left( 1 \right)^2 =18495 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 16^3 \\ & = (1+2+3+\cdots+16)^2 = 18496 \end{align*} \begin{align*} \therefore\quad 2^3 + 3^3 + 4^3+ \cdots+ 15^3 + 16^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 15^3 + 16^3\right) - \left( 1^3 \right) \\ &=18496 - \left( 1 \right)^2 =18495 \end{align*}
Find the value of \begin{equation*} 2^3 + 3^3 + 4^3+\cdots+16^3 + 17^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 17^3 \\ & = (1+2+3+\cdots+17)^2 = 23409 \end{align*} \begin{align*} \therefore\quad 2^3 + 3^3 + 4^3+ \cdots+ 16^3 + 17^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 16^3 + 17^3\right) - \left( 1^3 \right) \\ &=23409 - \left( 1 \right)^2 =23408 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 17^3 \\ & = (1+2+3+\cdots+17)^2 = 23409 \end{align*} \begin{align*} \therefore\quad 2^3 + 3^3 + 4^3+ \cdots+ 16^3 + 17^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 16^3 + 17^3\right) - \left( 1^3 \right) \\ &=23409 - \left( 1 \right)^2 =23408 \end{align*}
Find the value of \begin{equation*} 2^3 + 3^3 + 4^3+\cdots+17^3 + 18^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 18^3 \\ & = (1+2+3+\cdots+18)^2 = 29241 \end{align*} \begin{align*} \therefore\quad 2^3 + 3^3 + 4^3+ \cdots+ 17^3 + 18^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 17^3 + 18^3\right) - \left( 1^3 \right) \\ &=29241 - \left( 1 \right)^2 =29240 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 18^3 \\ & = (1+2+3+\cdots+18)^2 = 29241 \end{align*} \begin{align*} \therefore\quad 2^3 + 3^3 + 4^3+ \cdots+ 17^3 + 18^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 17^3 + 18^3\right) - \left( 1^3 \right) \\ &=29241 - \left( 1 \right)^2 =29240 \end{align*}
Find the value of \begin{equation*} 2^3 + 3^3 + 4^3+\cdots+18^3 + 19^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 19^3 \\ & = (1+2+3+\cdots+19)^2 = 36100 \end{align*} \begin{align*} \therefore\quad 2^3 + 3^3 + 4^3+ \cdots+ 18^3 + 19^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 18^3 + 19^3\right) - \left( 1^3 \right) \\ &=36100 - \left( 1 \right)^2 =36099 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 19^3 \\ & = (1+2+3+\cdots+19)^2 = 36100 \end{align*} \begin{align*} \therefore\quad 2^3 + 3^3 + 4^3+ \cdots+ 18^3 + 19^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 18^3 + 19^3\right) - \left( 1^3 \right) \\ &=36100 - \left( 1 \right)^2 =36099 \end{align*}
Find the value of \begin{equation*} 2^3 + 3^3 + 4^3+\cdots+19^3 + 20^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 20^3 \\ & = (1+2+3+\cdots+20)^2 = 44100 \end{align*} \begin{align*} \therefore\quad 2^3 + 3^3 + 4^3+ \cdots+ 19^3 + 20^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 19^3 + 20^3\right) - \left( 1^3 \right) \\ &=44100 - \left( 1 \right)^2 =44099 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 20^3 \\ & = (1+2+3+\cdots+20)^2 = 44100 \end{align*} \begin{align*} \therefore\quad 2^3 + 3^3 + 4^3+ \cdots+ 19^3 + 20^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 19^3 + 20^3\right) - \left( 1^3 \right) \\ &=44100 - \left( 1 \right)^2 =44099 \end{align*}
Find the value of \begin{equation*} 2^3 + 3^3 + 4^3+\cdots+20^3 + 21^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 21^3 \\ & = (1+2+3+\cdots+21)^2 = 53361 \end{align*} \begin{align*} \therefore\quad 2^3 + 3^3 + 4^3+ \cdots+ 20^3 + 21^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 20^3 + 21^3\right) - \left( 1^3 \right) \\ &=53361 - \left( 1 \right)^2 =53360 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 21^3 \\ & = (1+2+3+\cdots+21)^2 = 53361 \end{align*} \begin{align*} \therefore\quad 2^3 + 3^3 + 4^3+ \cdots+ 20^3 + 21^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 20^3 + 21^3\right) - \left( 1^3 \right) \\ &=53361 - \left( 1 \right)^2 =53360 \end{align*}
Find the value of \begin{equation*} 2^3 + 3^3 + 4^3+\cdots+21^3 + 22^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 22^3 \\ & = (1+2+3+\cdots+22)^2 = 64009 \end{align*} \begin{align*} \therefore\quad 2^3 + 3^3 + 4^3+ \cdots+ 21^3 + 22^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 21^3 + 22^3\right) - \left( 1^3 \right) \\ &=64009 - \left( 1 \right)^2 =64008 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 22^3 \\ & = (1+2+3+\cdots+22)^2 = 64009 \end{align*} \begin{align*} \therefore\quad 2^3 + 3^3 + 4^3+ \cdots+ 21^3 + 22^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 21^3 + 22^3\right) - \left( 1^3 \right) \\ &=64009 - \left( 1 \right)^2 =64008 \end{align*}
Find the value of \begin{equation*} 2^3 + 3^3 + 4^3+\cdots+22^3 + 23^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 23^3 \\ & = (1+2+3+\cdots+23)^2 = 76176 \end{align*} \begin{align*} \therefore\quad 2^3 + 3^3 + 4^3+ \cdots+ 22^3 + 23^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 22^3 + 23^3\right) - \left( 1^3 \right) \\ &=76176 - \left( 1 \right)^2 =76175 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 23^3 \\ & = (1+2+3+\cdots+23)^2 = 76176 \end{align*} \begin{align*} \therefore\quad 2^3 + 3^3 + 4^3+ \cdots+ 22^3 + 23^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 22^3 + 23^3\right) - \left( 1^3 \right) \\ &=76176 - \left( 1 \right)^2 =76175 \end{align*}
Find the value of \begin{equation*} 2^3 + 3^3 + 4^3+\cdots+23^3 + 24^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 24^3 \\ & = (1+2+3+\cdots+24)^2 = 90000 \end{align*} \begin{align*} \therefore\quad 2^3 + 3^3 + 4^3+ \cdots+ 23^3 + 24^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 23^3 + 24^3\right) - \left( 1^3 \right) \\ &=90000 - \left( 1 \right)^2 =89999 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 24^3 \\ & = (1+2+3+\cdots+24)^2 = 90000 \end{align*} \begin{align*} \therefore\quad 2^3 + 3^3 + 4^3+ \cdots+ 23^3 + 24^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 23^3 + 24^3\right) - \left( 1^3 \right) \\ &=90000 - \left( 1 \right)^2 =89999 \end{align*}
Find the value of \begin{equation*} 3^3 + 4^3 + 5^3+\cdots+13^3 + 14^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 14^3 \\ & = (1+2+3+\cdots+14)^2 = 11025 \end{align*} \begin{align*} \therefore\quad 3^3 + 4^3 + 5^3+ \cdots+ 13^3 + 14^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 13^3 + 14^3\right) - \left( 1^3 + 2^3 \right) \\ &=11025 - \left( 1 + 2 \right)^2 =11016 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 14^3 \\ & = (1+2+3+\cdots+14)^2 = 11025 \end{align*} \begin{align*} \therefore\quad 3^3 + 4^3 + 5^3+ \cdots+ 13^3 + 14^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 13^3 + 14^3\right) - \left( 1^3 + 2^3 \right) \\ &=11025 - \left( 1 + 2 \right)^2 =11016 \end{align*}
Find the value of \begin{equation*} 3^3 + 4^3 + 5^3+\cdots+14^3 + 15^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 15^3 \\ & = (1+2+3+\cdots+15)^2 = 14400 \end{align*} \begin{align*} \therefore\quad 3^3 + 4^3 + 5^3+ \cdots+ 14^3 + 15^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 14^3 + 15^3\right) - \left( 1^3 + 2^3 \right) \\ &=14400 - \left( 1 + 2 \right)^2 =14391 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 15^3 \\ & = (1+2+3+\cdots+15)^2 = 14400 \end{align*} \begin{align*} \therefore\quad 3^3 + 4^3 + 5^3+ \cdots+ 14^3 + 15^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 14^3 + 15^3\right) - \left( 1^3 + 2^3 \right) \\ &=14400 - \left( 1 + 2 \right)^2 =14391 \end{align*}
Find the value of \begin{equation*} 3^3 + 4^3 + 5^3+\cdots+15^3 + 16^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 16^3 \\ & = (1+2+3+\cdots+16)^2 = 18496 \end{align*} \begin{align*} \therefore\quad 3^3 + 4^3 + 5^3+ \cdots+ 15^3 + 16^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 15^3 + 16^3\right) - \left( 1^3 + 2^3 \right) \\ &=18496 - \left( 1 + 2 \right)^2 =18487 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 16^3 \\ & = (1+2+3+\cdots+16)^2 = 18496 \end{align*} \begin{align*} \therefore\quad 3^3 + 4^3 + 5^3+ \cdots+ 15^3 + 16^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 15^3 + 16^3\right) - \left( 1^3 + 2^3 \right) \\ &=18496 - \left( 1 + 2 \right)^2 =18487 \end{align*}
Find the value of \begin{equation*} 3^3 + 4^3 + 5^3+\cdots+16^3 + 17^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 17^3 \\ & = (1+2+3+\cdots+17)^2 = 23409 \end{align*} \begin{align*} \therefore\quad 3^3 + 4^3 + 5^3+ \cdots+ 16^3 + 17^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 16^3 + 17^3\right) - \left( 1^3 + 2^3 \right) \\ &=23409 - \left( 1 + 2 \right)^2 =23400 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 17^3 \\ & = (1+2+3+\cdots+17)^2 = 23409 \end{align*} \begin{align*} \therefore\quad 3^3 + 4^3 + 5^3+ \cdots+ 16^3 + 17^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 16^3 + 17^3\right) - \left( 1^3 + 2^3 \right) \\ &=23409 - \left( 1 + 2 \right)^2 =23400 \end{align*}
Find the value of \begin{equation*} 3^3 + 4^3 + 5^3+\cdots+17^3 + 18^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 18^3 \\ & = (1+2+3+\cdots+18)^2 = 29241 \end{align*} \begin{align*} \therefore\quad 3^3 + 4^3 + 5^3+ \cdots+ 17^3 + 18^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 17^3 + 18^3\right) - \left( 1^3 + 2^3 \right) \\ &=29241 - \left( 1 + 2 \right)^2 =29232 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 18^3 \\ & = (1+2+3+\cdots+18)^2 = 29241 \end{align*} \begin{align*} \therefore\quad 3^3 + 4^3 + 5^3+ \cdots+ 17^3 + 18^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 17^3 + 18^3\right) - \left( 1^3 + 2^3 \right) \\ &=29241 - \left( 1 + 2 \right)^2 =29232 \end{align*}
Find the value of \begin{equation*} 3^3 + 4^3 + 5^3+\cdots+18^3 + 19^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 19^3 \\ & = (1+2+3+\cdots+19)^2 = 36100 \end{align*} \begin{align*} \therefore\quad 3^3 + 4^3 + 5^3+ \cdots+ 18^3 + 19^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 18^3 + 19^3\right) - \left( 1^3 + 2^3 \right) \\ &=36100 - \left( 1 + 2 \right)^2 =36091 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 19^3 \\ & = (1+2+3+\cdots+19)^2 = 36100 \end{align*} \begin{align*} \therefore\quad 3^3 + 4^3 + 5^3+ \cdots+ 18^3 + 19^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 18^3 + 19^3\right) - \left( 1^3 + 2^3 \right) \\ &=36100 - \left( 1 + 2 \right)^2 =36091 \end{align*}
Find the value of \begin{equation*} 3^3 + 4^3 + 5^3+\cdots+19^3 + 20^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 20^3 \\ & = (1+2+3+\cdots+20)^2 = 44100 \end{align*} \begin{align*} \therefore\quad 3^3 + 4^3 + 5^3+ \cdots+ 19^3 + 20^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 19^3 + 20^3\right) - \left( 1^3 + 2^3 \right) \\ &=44100 - \left( 1 + 2 \right)^2 =44091 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 20^3 \\ & = (1+2+3+\cdots+20)^2 = 44100 \end{align*} \begin{align*} \therefore\quad 3^3 + 4^3 + 5^3+ \cdots+ 19^3 + 20^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 19^3 + 20^3\right) - \left( 1^3 + 2^3 \right) \\ &=44100 - \left( 1 + 2 \right)^2 =44091 \end{align*}
Find the value of \begin{equation*} 3^3 + 4^3 + 5^3+\cdots+20^3 + 21^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 21^3 \\ & = (1+2+3+\cdots+21)^2 = 53361 \end{align*} \begin{align*} \therefore\quad 3^3 + 4^3 + 5^3+ \cdots+ 20^3 + 21^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 20^3 + 21^3\right) - \left( 1^3 + 2^3 \right) \\ &=53361 - \left( 1 + 2 \right)^2 =53352 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 21^3 \\ & = (1+2+3+\cdots+21)^2 = 53361 \end{align*} \begin{align*} \therefore\quad 3^3 + 4^3 + 5^3+ \cdots+ 20^3 + 21^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 20^3 + 21^3\right) - \left( 1^3 + 2^3 \right) \\ &=53361 - \left( 1 + 2 \right)^2 =53352 \end{align*}
Find the value of \begin{equation*} 3^3 + 4^3 + 5^3+\cdots+21^3 + 22^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 22^3 \\ & = (1+2+3+\cdots+22)^2 = 64009 \end{align*} \begin{align*} \therefore\quad 3^3 + 4^3 + 5^3+ \cdots+ 21^3 + 22^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 21^3 + 22^3\right) - \left( 1^3 + 2^3 \right) \\ &=64009 - \left( 1 + 2 \right)^2 =64000 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 22^3 \\ & = (1+2+3+\cdots+22)^2 = 64009 \end{align*} \begin{align*} \therefore\quad 3^3 + 4^3 + 5^3+ \cdots+ 21^3 + 22^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 21^3 + 22^3\right) - \left( 1^3 + 2^3 \right) \\ &=64009 - \left( 1 + 2 \right)^2 =64000 \end{align*}
Find the value of \begin{equation*} 3^3 + 4^3 + 5^3+\cdots+22^3 + 23^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 23^3 \\ & = (1+2+3+\cdots+23)^2 = 76176 \end{align*} \begin{align*} \therefore\quad 3^3 + 4^3 + 5^3+ \cdots+ 22^3 + 23^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 22^3 + 23^3\right) - \left( 1^3 + 2^3 \right) \\ &=76176 - \left( 1 + 2 \right)^2 =76167 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 23^3 \\ & = (1+2+3+\cdots+23)^2 = 76176 \end{align*} \begin{align*} \therefore\quad 3^3 + 4^3 + 5^3+ \cdots+ 22^3 + 23^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 22^3 + 23^3\right) - \left( 1^3 + 2^3 \right) \\ &=76176 - \left( 1 + 2 \right)^2 =76167 \end{align*}
Find the value of \begin{equation*} 3^3 + 4^3 + 5^3+\cdots+23^3 + 24^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 24^3 \\ & = (1+2+3+\cdots+24)^2 = 90000 \end{align*} \begin{align*} \therefore\quad 3^3 + 4^3 + 5^3+ \cdots+ 23^3 + 24^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 23^3 + 24^3\right) - \left( 1^3 + 2^3 \right) \\ &=90000 - \left( 1 + 2 \right)^2 =89991 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 24^3 \\ & = (1+2+3+\cdots+24)^2 = 90000 \end{align*} \begin{align*} \therefore\quad 3^3 + 4^3 + 5^3+ \cdots+ 23^3 + 24^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 23^3 + 24^3\right) - \left( 1^3 + 2^3 \right) \\ &=90000 - \left( 1 + 2 \right)^2 =89991 \end{align*}
Find the value of \begin{equation*} 4^3 + 5^3 + 6^3+\cdots+13^3 + 14^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 14^3 \\ & = (1+2+3+\cdots+14)^2 = 11025 \end{align*} \begin{align*} \therefore\quad 4^3 + 5^3 + 6^3+ \cdots+ 13^3 + 14^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 13^3 + 14^3\right) - \left( 1^3 + 2^3 + 3^3 \right) \\ &=11025 - \left( 1 + 2 + 3 \right)^2 =10989 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 14^3 \\ & = (1+2+3+\cdots+14)^2 = 11025 \end{align*} \begin{align*} \therefore\quad 4^3 + 5^3 + 6^3+ \cdots+ 13^3 + 14^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 13^3 + 14^3\right) - \left( 1^3 + 2^3 + 3^3 \right) \\ &=11025 - \left( 1 + 2 + 3 \right)^2 =10989 \end{align*}
Find the value of \begin{equation*} 4^3 + 5^3 + 6^3+\cdots+14^3 + 15^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 15^3 \\ & = (1+2+3+\cdots+15)^2 = 14400 \end{align*} \begin{align*} \therefore\quad 4^3 + 5^3 + 6^3+ \cdots+ 14^3 + 15^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 14^3 + 15^3\right) - \left( 1^3 + 2^3 + 3^3 \right) \\ &=14400 - \left( 1 + 2 + 3 \right)^2 =14364 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 15^3 \\ & = (1+2+3+\cdots+15)^2 = 14400 \end{align*} \begin{align*} \therefore\quad 4^3 + 5^3 + 6^3+ \cdots+ 14^3 + 15^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 14^3 + 15^3\right) - \left( 1^3 + 2^3 + 3^3 \right) \\ &=14400 - \left( 1 + 2 + 3 \right)^2 =14364 \end{align*}
Find the value of \begin{equation*} 4^3 + 5^3 + 6^3+\cdots+15^3 + 16^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 16^3 \\ & = (1+2+3+\cdots+16)^2 = 18496 \end{align*} \begin{align*} \therefore\quad 4^3 + 5^3 + 6^3+ \cdots+ 15^3 + 16^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 15^3 + 16^3\right) - \left( 1^3 + 2^3 + 3^3 \right) \\ &=18496 - \left( 1 + 2 + 3 \right)^2 =18460 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 16^3 \\ & = (1+2+3+\cdots+16)^2 = 18496 \end{align*} \begin{align*} \therefore\quad 4^3 + 5^3 + 6^3+ \cdots+ 15^3 + 16^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 15^3 + 16^3\right) - \left( 1^3 + 2^3 + 3^3 \right) \\ &=18496 - \left( 1 + 2 + 3 \right)^2 =18460 \end{align*}
Find the value of \begin{equation*} 4^3 + 5^3 + 6^3+\cdots+16^3 + 17^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 17^3 \\ & = (1+2+3+\cdots+17)^2 = 23409 \end{align*} \begin{align*} \therefore\quad 4^3 + 5^3 + 6^3+ \cdots+ 16^3 + 17^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 16^3 + 17^3\right) - \left( 1^3 + 2^3 + 3^3 \right) \\ &=23409 - \left( 1 + 2 + 3 \right)^2 =23373 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 17^3 \\ & = (1+2+3+\cdots+17)^2 = 23409 \end{align*} \begin{align*} \therefore\quad 4^3 + 5^3 + 6^3+ \cdots+ 16^3 + 17^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 16^3 + 17^3\right) - \left( 1^3 + 2^3 + 3^3 \right) \\ &=23409 - \left( 1 + 2 + 3 \right)^2 =23373 \end{align*}
Find the value of \begin{equation*} 4^3 + 5^3 + 6^3+\cdots+17^3 + 18^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 18^3 \\ & = (1+2+3+\cdots+18)^2 = 29241 \end{align*} \begin{align*} \therefore\quad 4^3 + 5^3 + 6^3+ \cdots+ 17^3 + 18^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 17^3 + 18^3\right) - \left( 1^3 + 2^3 + 3^3 \right) \\ &=29241 - \left( 1 + 2 + 3 \right)^2 =29205 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 18^3 \\ & = (1+2+3+\cdots+18)^2 = 29241 \end{align*} \begin{align*} \therefore\quad 4^3 + 5^3 + 6^3+ \cdots+ 17^3 + 18^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 17^3 + 18^3\right) - \left( 1^3 + 2^3 + 3^3 \right) \\ &=29241 - \left( 1 + 2 + 3 \right)^2 =29205 \end{align*}
Find the value of \begin{equation*} 4^3 + 5^3 + 6^3+\cdots+18^3 + 19^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 19^3 \\ & = (1+2+3+\cdots+19)^2 = 36100 \end{align*} \begin{align*} \therefore\quad 4^3 + 5^3 + 6^3+ \cdots+ 18^3 + 19^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 18^3 + 19^3\right) - \left( 1^3 + 2^3 + 3^3 \right) \\ &=36100 - \left( 1 + 2 + 3 \right)^2 =36064 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 19^3 \\ & = (1+2+3+\cdots+19)^2 = 36100 \end{align*} \begin{align*} \therefore\quad 4^3 + 5^3 + 6^3+ \cdots+ 18^3 + 19^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 18^3 + 19^3\right) - \left( 1^3 + 2^3 + 3^3 \right) \\ &=36100 - \left( 1 + 2 + 3 \right)^2 =36064 \end{align*}
Find the value of \begin{equation*} 4^3 + 5^3 + 6^3+\cdots+19^3 + 20^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 20^3 \\ & = (1+2+3+\cdots+20)^2 = 44100 \end{align*} \begin{align*} \therefore\quad 4^3 + 5^3 + 6^3+ \cdots+ 19^3 + 20^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 19^3 + 20^3\right) - \left( 1^3 + 2^3 + 3^3 \right) \\ &=44100 - \left( 1 + 2 + 3 \right)^2 =44064 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 20^3 \\ & = (1+2+3+\cdots+20)^2 = 44100 \end{align*} \begin{align*} \therefore\quad 4^3 + 5^3 + 6^3+ \cdots+ 19^3 + 20^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 19^3 + 20^3\right) - \left( 1^3 + 2^3 + 3^3 \right) \\ &=44100 - \left( 1 + 2 + 3 \right)^2 =44064 \end{align*}
Find the value of \begin{equation*} 4^3 + 5^3 + 6^3+\cdots+20^3 + 21^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 21^3 \\ & = (1+2+3+\cdots+21)^2 = 53361 \end{align*} \begin{align*} \therefore\quad 4^3 + 5^3 + 6^3+ \cdots+ 20^3 + 21^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 20^3 + 21^3\right) - \left( 1^3 + 2^3 + 3^3 \right) \\ &=53361 - \left( 1 + 2 + 3 \right)^2 =53325 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 21^3 \\ & = (1+2+3+\cdots+21)^2 = 53361 \end{align*} \begin{align*} \therefore\quad 4^3 + 5^3 + 6^3+ \cdots+ 20^3 + 21^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 20^3 + 21^3\right) - \left( 1^3 + 2^3 + 3^3 \right) \\ &=53361 - \left( 1 + 2 + 3 \right)^2 =53325 \end{align*}
Find the value of \begin{equation*} 4^3 + 5^3 + 6^3+\cdots+21^3 + 22^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 22^3 \\ & = (1+2+3+\cdots+22)^2 = 64009 \end{align*} \begin{align*} \therefore\quad 4^3 + 5^3 + 6^3+ \cdots+ 21^3 + 22^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 21^3 + 22^3\right) - \left( 1^3 + 2^3 + 3^3 \right) \\ &=64009 - \left( 1 + 2 + 3 \right)^2 =63973 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 22^3 \\ & = (1+2+3+\cdots+22)^2 = 64009 \end{align*} \begin{align*} \therefore\quad 4^3 + 5^3 + 6^3+ \cdots+ 21^3 + 22^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 21^3 + 22^3\right) - \left( 1^3 + 2^3 + 3^3 \right) \\ &=64009 - \left( 1 + 2 + 3 \right)^2 =63973 \end{align*}
Find the value of \begin{equation*} 4^3 + 5^3 + 6^3+\cdots+22^3 + 23^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 23^3 \\ & = (1+2+3+\cdots+23)^2 = 76176 \end{align*} \begin{align*} \therefore\quad 4^3 + 5^3 + 6^3+ \cdots+ 22^3 + 23^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 22^3 + 23^3\right) - \left( 1^3 + 2^3 + 3^3 \right) \\ &=76176 - \left( 1 + 2 + 3 \right)^2 =76140 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 23^3 \\ & = (1+2+3+\cdots+23)^2 = 76176 \end{align*} \begin{align*} \therefore\quad 4^3 + 5^3 + 6^3+ \cdots+ 22^3 + 23^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 22^3 + 23^3\right) - \left( 1^3 + 2^3 + 3^3 \right) \\ &=76176 - \left( 1 + 2 + 3 \right)^2 =76140 \end{align*}
Find the value of \begin{equation*} 4^3 + 5^3 + 6^3+\cdots+23^3 + 24^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 24^3 \\ & = (1+2+3+\cdots+24)^2 = 90000 \end{align*} \begin{align*} \therefore\quad 4^3 + 5^3 + 6^3+ \cdots+ 23^3 + 24^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 23^3 + 24^3\right) - \left( 1^3 + 2^3 + 3^3 \right) \\ &=90000 - \left( 1 + 2 + 3 \right)^2 =89964 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 24^3 \\ & = (1+2+3+\cdots+24)^2 = 90000 \end{align*} \begin{align*} \therefore\quad 4^3 + 5^3 + 6^3+ \cdots+ 23^3 + 24^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 23^3 + 24^3\right) - \left( 1^3 + 2^3 + 3^3 \right) \\ &=90000 - \left( 1 + 2 + 3 \right)^2 =89964 \end{align*}
Find the value of \begin{equation*} 5^3 + 6^3 + 7^3+\cdots+13^3 + 14^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 14^3 \\ & = (1+2+3+\cdots+14)^2 = 11025 \end{align*} \begin{align*} \therefore\quad 5^3 + 6^3 + 7^3+ \cdots+ 13^3 + 14^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 13^3 + 14^3\right) - \left( 1^3 + 2^3 + 3^3 + 4^3 \right) \\ &=11025 - \left( 1 + 2 + 3 + 4 \right)^2 =10925 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 14^3 \\ & = (1+2+3+\cdots+14)^2 = 11025 \end{align*} \begin{align*} \therefore\quad 5^3 + 6^3 + 7^3+ \cdots+ 13^3 + 14^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 13^3 + 14^3\right) - \left( 1^3 + 2^3 + 3^3 + 4^3 \right) \\ &=11025 - \left( 1 + 2 + 3 + 4 \right)^2 =10925 \end{align*}
Find the value of \begin{equation*} 5^3 + 6^3 + 7^3+\cdots+14^3 + 15^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 15^3 \\ & = (1+2+3+\cdots+15)^2 = 14400 \end{align*} \begin{align*} \therefore\quad 5^3 + 6^3 + 7^3+ \cdots+ 14^3 + 15^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 14^3 + 15^3\right) - \left( 1^3 + 2^3 + 3^3 + 4^3 \right) \\ &=14400 - \left( 1 + 2 + 3 + 4 \right)^2 =14300 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 15^3 \\ & = (1+2+3+\cdots+15)^2 = 14400 \end{align*} \begin{align*} \therefore\quad 5^3 + 6^3 + 7^3+ \cdots+ 14^3 + 15^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 14^3 + 15^3\right) - \left( 1^3 + 2^3 + 3^3 + 4^3 \right) \\ &=14400 - \left( 1 + 2 + 3 + 4 \right)^2 =14300 \end{align*}
Find the value of \begin{equation*} 5^3 + 6^3 + 7^3+\cdots+15^3 + 16^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 16^3 \\ & = (1+2+3+\cdots+16)^2 = 18496 \end{align*} \begin{align*} \therefore\quad 5^3 + 6^3 + 7^3+ \cdots+ 15^3 + 16^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 15^3 + 16^3\right) - \left( 1^3 + 2^3 + 3^3 + 4^3 \right) \\ &=18496 - \left( 1 + 2 + 3 + 4 \right)^2 =18396 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 16^3 \\ & = (1+2+3+\cdots+16)^2 = 18496 \end{align*} \begin{align*} \therefore\quad 5^3 + 6^3 + 7^3+ \cdots+ 15^3 + 16^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 15^3 + 16^3\right) - \left( 1^3 + 2^3 + 3^3 + 4^3 \right) \\ &=18496 - \left( 1 + 2 + 3 + 4 \right)^2 =18396 \end{align*}
Find the value of \begin{equation*} 5^3 + 6^3 + 7^3+\cdots+16^3 + 17^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 17^3 \\ & = (1+2+3+\cdots+17)^2 = 23409 \end{align*} \begin{align*} \therefore\quad 5^3 + 6^3 + 7^3+ \cdots+ 16^3 + 17^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 16^3 + 17^3\right) - \left( 1^3 + 2^3 + 3^3 + 4^3 \right) \\ &=23409 - \left( 1 + 2 + 3 + 4 \right)^2 =23309 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 17^3 \\ & = (1+2+3+\cdots+17)^2 = 23409 \end{align*} \begin{align*} \therefore\quad 5^3 + 6^3 + 7^3+ \cdots+ 16^3 + 17^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 16^3 + 17^3\right) - \left( 1^3 + 2^3 + 3^3 + 4^3 \right) \\ &=23409 - \left( 1 + 2 + 3 + 4 \right)^2 =23309 \end{align*}
Find the value of \begin{equation*} 5^3 + 6^3 + 7^3+\cdots+17^3 + 18^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 18^3 \\ & = (1+2+3+\cdots+18)^2 = 29241 \end{align*} \begin{align*} \therefore\quad 5^3 + 6^3 + 7^3+ \cdots+ 17^3 + 18^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 17^3 + 18^3\right) - \left( 1^3 + 2^3 + 3^3 + 4^3 \right) \\ &=29241 - \left( 1 + 2 + 3 + 4 \right)^2 =29141 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 18^3 \\ & = (1+2+3+\cdots+18)^2 = 29241 \end{align*} \begin{align*} \therefore\quad 5^3 + 6^3 + 7^3+ \cdots+ 17^3 + 18^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 17^3 + 18^3\right) - \left( 1^3 + 2^3 + 3^3 + 4^3 \right) \\ &=29241 - \left( 1 + 2 + 3 + 4 \right)^2 =29141 \end{align*}
Find the value of \begin{equation*} 5^3 + 6^3 + 7^3+\cdots+18^3 + 19^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 19^3 \\ & = (1+2+3+\cdots+19)^2 = 36100 \end{align*} \begin{align*} \therefore\quad 5^3 + 6^3 + 7^3+ \cdots+ 18^3 + 19^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 18^3 + 19^3\right) - \left( 1^3 + 2^3 + 3^3 + 4^3 \right) \\ &=36100 - \left( 1 + 2 + 3 + 4 \right)^2 =36000 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 19^3 \\ & = (1+2+3+\cdots+19)^2 = 36100 \end{align*} \begin{align*} \therefore\quad 5^3 + 6^3 + 7^3+ \cdots+ 18^3 + 19^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 18^3 + 19^3\right) - \left( 1^3 + 2^3 + 3^3 + 4^3 \right) \\ &=36100 - \left( 1 + 2 + 3 + 4 \right)^2 =36000 \end{align*}
Find the value of \begin{equation*} 5^3 + 6^3 + 7^3+\cdots+19^3 + 20^3 \end{equation*}
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\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 20^3 \\ & = (1+2+3+\cdots+20)^2 = 44100 \end{align*} \begin{align*} \therefore\quad 5^3 + 6^3 + 7^3+ \cdots+ 19^3 + 20^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 19^3 + 20^3\right) - \left( 1^3 + 2^3 + 3^3 + 4^3 \right) \\ &=44100 - \left( 1 + 2 + 3 + 4 \right)^2 =44000 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 20^3 \\ & = (1+2+3+\cdots+20)^2 = 44100 \end{align*} \begin{align*} \therefore\quad 5^3 + 6^3 + 7^3+ \cdots+ 19^3 + 20^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 19^3 + 20^3\right) - \left( 1^3 + 2^3 + 3^3 + 4^3 \right) \\ &=44100 - \left( 1 + 2 + 3 + 4 \right)^2 =44000 \end{align*}
Find the value of \begin{equation*} 5^3 + 6^3 + 7^3+\cdots+20^3 + 21^3 \end{equation*}
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Sorry. Please check the correct answer below.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 21^3 \\ & = (1+2+3+\cdots+21)^2 = 53361 \end{align*} \begin{align*} \therefore\quad 5^3 + 6^3 + 7^3+ \cdots+ 20^3 + 21^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 20^3 + 21^3\right) - \left( 1^3 + 2^3 + 3^3 + 4^3 \right) \\ &=53361 - \left( 1 + 2 + 3 + 4 \right)^2 =53261 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 21^3 \\ & = (1+2+3+\cdots+21)^2 = 53361 \end{align*} \begin{align*} \therefore\quad 5^3 + 6^3 + 7^3+ \cdots+ 20^3 + 21^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 20^3 + 21^3\right) - \left( 1^3 + 2^3 + 3^3 + 4^3 \right) \\ &=53361 - \left( 1 + 2 + 3 + 4 \right)^2 =53261 \end{align*}
Find the value of \begin{equation*} 5^3 + 6^3 + 7^3+\cdots+21^3 + 22^3 \end{equation*}
Select An Answer
Sorry. Please check the correct answer below.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 22^3 \\ & = (1+2+3+\cdots+22)^2 = 64009 \end{align*} \begin{align*} \therefore\quad 5^3 + 6^3 + 7^3+ \cdots+ 21^3 + 22^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 21^3 + 22^3\right) - \left( 1^3 + 2^3 + 3^3 + 4^3 \right) \\ &=64009 - \left( 1 + 2 + 3 + 4 \right)^2 =63909 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 22^3 \\ & = (1+2+3+\cdots+22)^2 = 64009 \end{align*} \begin{align*} \therefore\quad 5^3 + 6^3 + 7^3+ \cdots+ 21^3 + 22^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 21^3 + 22^3\right) - \left( 1^3 + 2^3 + 3^3 + 4^3 \right) \\ &=64009 - \left( 1 + 2 + 3 + 4 \right)^2 =63909 \end{align*}
Find the value of \begin{equation*} 5^3 + 6^3 + 7^3+\cdots+22^3 + 23^3 \end{equation*}
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Sorry. Please check the correct answer below.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 23^3 \\ & = (1+2+3+\cdots+23)^2 = 76176 \end{align*} \begin{align*} \therefore\quad 5^3 + 6^3 + 7^3+ \cdots+ 22^3 + 23^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 22^3 + 23^3\right) - \left( 1^3 + 2^3 + 3^3 + 4^3 \right) \\ &=76176 - \left( 1 + 2 + 3 + 4 \right)^2 =76076 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 23^3 \\ & = (1+2+3+\cdots+23)^2 = 76176 \end{align*} \begin{align*} \therefore\quad 5^3 + 6^3 + 7^3+ \cdots+ 22^3 + 23^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 22^3 + 23^3\right) - \left( 1^3 + 2^3 + 3^3 + 4^3 \right) \\ &=76176 - \left( 1 + 2 + 3 + 4 \right)^2 =76076 \end{align*}
Find the value of \begin{equation*} 5^3 + 6^3 + 7^3+\cdots+23^3 + 24^3 \end{equation*}
Select An Answer
Sorry. Please check the correct answer below.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 24^3 \\ & = (1+2+3+\cdots+24)^2 = 90000 \end{align*} \begin{align*} \therefore\quad 5^3 + 6^3 + 7^3+ \cdots+ 23^3 + 24^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 23^3 + 24^3\right) - \left( 1^3 + 2^3 + 3^3 + 4^3 \right) \\ &=90000 - \left( 1 + 2 + 3 + 4 \right)^2 =89900 \end{align*}
Yay! Your are right.
\begin{align*} \text{Chose} = 1^3+2^3 &=1+8=9 \\ 1^3+2^3+3^3 &= 9+27 =36 \\ (1+2)^2 &= 9 \\ (1+2+3)^2 &= 36 \end{align*} \begin{align*} \therefore\quad & 1^3+2^3+3^3+ \cdots+ 24^3 \\ & = (1+2+3+\cdots+24)^2 = 90000 \end{align*} \begin{align*} \therefore\quad 5^3 + 6^3 + 7^3+ \cdots+ 23^3 + 24^3 &= \left(1^3 + 2^3 + 3^3+ \cdots+ 23^3 + 24^3\right) - \left( 1^3 + 2^3 + 3^3 + 4^3 \right) \\ &=90000 - \left( 1 + 2 + 3 + 4 \right)^2 =89900 \end{align*}
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