Question 1 of 213

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\begin{equation*} 1+(-2+3)+(-4+5)+\cdots+(-2016+2017) \end{equation*} \begin{align*} -2+3 &= 3-2 = 1\\ -4+5 &= 5-4 = 1\\ & \hspace{2.5mm} \vdots \\ \ -2016 &+ 2017 = 1 \end{align*} \begin{equation*} (2016\div 2)\times 1 + 1 = 1009 \end{equation*}

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Assume all to be $\frac{1}{2008}$ \begin{equation} \frac{1}{\frac{10}{2008}} = 200.8 \end{equation} assume all to be $\frac{1}{2017}$ \begin{equation} \frac{1}{\frac{10}{2017}}=201.7 \end{equation} \begin{equation} 200.8 < S < 201.7 \end{equation} $c=7$.

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\begin{equation*} 1+(-2+3)+(-4+5)+\cdots+(-1908+1909) \end{equation*} \begin{align*} -2+3 &= 3-2 = 1 \\ -4+5 &= 5-4 = 1 \\ & \hspace{2.5mm} \vdots \\ \ -1908 &+ 1909 = 1 \end{align*} \begin{equation*} (1908\div 2)\times 1 + 1 = 955 \end{equation*}

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\begin{equation*} 1+(-2+3)+(-4+5)+\cdots+(-1916+1917) \end{equation*} \begin{align*} -2+3 &= 3-2 = 1 \\ -4+5 &= 5-4 = 1 \\ & \hspace{2.5mm} \vdots \\ \ -1916 &+ 1917 = 1 \end{align*} \begin{equation*} (1916\div 2)\times 1 + 1 = 959 \end{equation*}

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\begin{equation*} 1+(-2+3)+(-4+5)+\cdots+(-1918+1919) \end{equation*} \begin{align*} -2+3 &= 3-2 = 1 \\ -4+5 &= 5-4 = 1 \\ & \hspace{2.5mm} \vdots \\ \ -1918 &+ 1919 = 1 \end{align*} \begin{equation*} (1918\div 2)\times 1 + 1 = 960 \end{equation*}

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\begin{equation*} 1+(-2+3)+(-4+5)+\cdots+(-1938+1939) \end{equation*} \begin{align*} -2+3 &= 3-2 = 1 \\ -4+5 &= 5-4 = 1 \\ & \hspace{2.5mm} \vdots \\ \ -1938 &+ 1939 = 1 \end{align*} \begin{equation*} (1938\div 2)\times 1 + 1 = 970 \end{equation*}

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\begin{equation*} 1+(-2+3)+(-4+5)+\cdots+(-1958+1959) \end{equation*} \begin{align*} -2+3 &= 3-2 = 1 \\ -4+5 &= 5-4 = 1 \\ & \hspace{2.5mm} \vdots \\ \ -1958 &+ 1959 = 1 \end{align*} \begin{equation*} (1958\div 2)\times 1 + 1 = 980 \end{equation*}

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\begin{equation*} 1+(-2+3)+(-4+5)+\cdots+(-1998+1999) \end{equation*} \begin{align*} -2+3 &= 3-2 = 1 \\ -4+5 &= 5-4 = 1 \\ & \hspace{2.5mm} \vdots \\ \ -1998 &+ 1999 = 1 \end{align*} \begin{equation*} (1998\div 2)\times 1 + 1 = 1000 \end{equation*}

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\begin{equation*} 1+(-2+3)+(-4+5)+\cdots+(-2020+2021) \end{equation*} \begin{align*} -2+3 &= 3-2 = 1 \\ -4+5 &= 5-4 = 1 \\ & \hspace{2.5mm} \vdots \\ \ -2020 &+ 2021 = 1 \end{align*} \begin{equation*} (2020\div 2)\times 1 + 1 = 1011 \end{equation*}

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\begin{equation*} 1+(-2+3)+(-4+5)+\cdots+(-2040+2041) \end{equation*} \begin{align*} -2+3 &= 3-2 = 1 \\ -4+5 &= 5-4 = 1 \\ & \hspace{2.5mm} \vdots \\ \ -2040 &+ 2041 = 1 \end{align*} \begin{equation*} (2040\div 2)\times 1 + 1 = 1021 \end{equation*}

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\begin{equation*} 1+(-2+3)+(-4+5)+\cdots+(-2076+2077) \end{equation*} \begin{align*} -2+3 &= 3-2 = 1 \\ -4+5 &= 5-4 = 1 \\ & \hspace{2.5mm} \vdots \\ \ -2076 &+ 2077 = 1 \end{align*} \begin{equation*} (2076\div 2)\times 1 + 1 = 1039 \end{equation*}

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\begin{equation*} 1+(-2+3)+(-4+5)+\cdots+(-2098+2099) \end{equation*} \begin{align*} -2+3 &= 3-2 = 1 \\ -4+5 &= 5-4 = 1 \\ & \hspace{2.5mm} \vdots \\ \ -2098 &+ 2099 = 1 \end{align*} \begin{equation*} (2098\div 2)\times 1 + 1 = 1050 \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2251}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2250}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2251} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1645}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1644}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1645} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1978}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1977}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1978} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2204}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2203}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2204} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2027}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2026}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2027} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1874}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1873}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1874} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2139}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2138}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2139} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1694}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1693}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1694} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1976}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1975}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1976} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1611}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1610}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1611} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2062}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2061}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2062} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1612}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1611}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1612} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1779}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1778}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1779} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1988}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1987}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1988} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1984}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1983}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1984} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2290}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2289}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2290} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1589}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1588}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1589} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2167}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2166}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2167} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1666}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1665}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1666} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2352}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2351}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2352} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1990}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1989}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1990} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1722}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1721}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1722} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2012}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2011}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2012} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1643}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1642}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1643} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2116}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2115}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2116} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1688}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1687}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1688} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1629}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1628}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1629} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1527}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1526}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1527} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2032}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2031}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2032} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1603}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1602}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1603} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1606}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1605}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1606} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2114}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2113}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2114} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1902}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1901}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1902} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2298}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2297}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2298} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2261}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2260}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2261} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1641}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1640}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1641} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2157}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2156}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2157} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2368}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2367}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2368} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2438}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2437}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2438} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2388}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2387}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2388} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2193}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2192}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2193} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1734}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1733}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1734} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2022}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2021}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2022} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1806}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1805}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1806} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1812}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1811}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1812} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2430}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2429}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2430} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2396}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2395}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2396} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1994}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1993}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1994} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1530}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1529}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1530} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1534}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1533}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1534} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2478}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2477}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2478} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1909}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1908}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1909} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2454}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2453}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2454} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1599}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1598}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1599} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2013}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2012}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2013} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1510}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1509}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1510} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2232}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2231}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2232} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2151}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2150}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2151} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2023}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2022}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2023} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2059}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2058}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2059} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2257}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2256}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2257} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2351}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2350}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2351} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1549}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1548}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1549} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1928}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1927}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1928} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1841}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1840}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1841} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1833}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1832}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1833} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2300}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2299}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2300} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1720}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1719}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1720} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2120}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2119}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2120} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2399}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2398}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2399} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1685}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1684}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1685} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2390}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2389}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2390} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1538}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1537}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1538} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1888}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1887}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1888} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1512}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1511}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1512} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2408}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2407}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2408} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2288}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2287}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2288} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1824}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1823}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1824} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1713}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1712}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1713} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2452}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2451}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2452} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2104}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2103}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2104} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2064}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2063}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2064} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1838}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1837}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1838} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2244}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2243}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2244} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1896}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1895}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1896} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2277}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2276}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2277} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2303}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2302}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2303} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1665}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1664}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1665} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1802}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1801}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1802} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2359}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2358}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2359} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1869}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1868}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1869} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1532}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1531}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1532} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1742}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1741}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1742} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1562}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1561}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1562} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1889}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1888}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1889} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2289}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2288}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2289} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2225}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2224}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2225} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2412}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2411}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2412} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2037}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{2036}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{2037} \end{equation*}

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\begin{align*} \text{Let } A&= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1644}\right) \\ B &= \left(\tfrac{1}{2}+\tfrac{1}{3}+\cdots+\tfrac{1}{1643}\right) \end{align*} \begin{equation*} A\times (1+B)-(1+A)\times B = \tfrac{1}{1644} \end{equation*}

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Assume all to be $\frac{1}{2000}$ \begin{equation} \frac{1}{\frac{7}{2000}} = 285.7 \end{equation} assume all to be $\frac{1}{2006}$ \begin{equation} \frac{1}{\frac{7}{2006}}= 286.6 \end{equation} \begin{equation} 285.7 < S < 286.6 \end{equation}

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Assume all to be $\frac{1}{2000}$ \begin{equation} \frac{1}{\frac{8}{2000}} = 250.0 \end{equation} assume all to be $\frac{1}{2007}$ \begin{equation} \frac{1}{\frac{8}{2007}}= 250.9 \end{equation} \begin{equation} 250.0 < S < 250.9 \end{equation}

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Assume all to be $\frac{1}{2000}$ \begin{equation} \frac{1}{\frac{10}{2000}} = 200.0 \end{equation} assume all to be $\frac{1}{2009}$ \begin{equation} \frac{1}{\frac{10}{2009}}= 200.9 \end{equation} \begin{equation} 200.0 < S < 200.9 \end{equation}

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Assume all to be $\frac{1}{2000}$ \begin{equation} \frac{1}{\frac{11}{2000}} = 181.8 \end{equation} assume all to be $\frac{1}{2010}$ \begin{equation} \frac{1}{\frac{11}{2010}}= 182.7 \end{equation} \begin{equation} 181.8 < S < 182.7 \end{equation}

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Assume all to be $\frac{1}{2000}$ \begin{equation} \frac{1}{\frac{12}{2000}} = 166.7 \end{equation} assume all to be $\frac{1}{2011}$ \begin{equation} \frac{1}{\frac{12}{2011}}= 167.6 \end{equation} \begin{equation} 166.7 < S < 167.6 \end{equation}

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Assume all to be $\frac{1}{2000}$ \begin{equation} \frac{1}{\frac{13}{2000}} = 153.8 \end{equation} assume all to be $\frac{1}{2012}$ \begin{equation} \frac{1}{\frac{13}{2012}}= 154.8 \end{equation} \begin{equation} 153.8 < S < 154.8 \end{equation}

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Assume all to be $\frac{1}{2000}$ \begin{equation} \frac{1}{\frac{14}{2000}} = 142.9 \end{equation} assume all to be $\frac{1}{2013}$ \begin{equation} \frac{1}{\frac{14}{2013}}= 143.8 \end{equation} \begin{equation} 142.9 < S < 143.8 \end{equation}

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Assume all to be $\frac{1}{2000}$ \begin{equation} \frac{1}{\frac{16}{2000}} = 125.0 \end{equation} assume all to be $\frac{1}{2015}$ \begin{equation} \frac{1}{\frac{16}{2015}}= 125.9 \end{equation} \begin{equation} 125.0 < S < 125.9 \end{equation}

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Assume all to be $\frac{1}{2000}$ \begin{equation} \frac{1}{\frac{17}{2000}} = 117.6 \end{equation} assume all to be $\frac{1}{2016}$ \begin{equation} \frac{1}{\frac{17}{2016}}= 118.6 \end{equation} \begin{equation} 117.6 < S < 118.6 \end{equation}

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Assume all to be $\frac{1}{2000}$ \begin{equation} \frac{1}{\frac{20}{2000}} = 100.0 \end{equation} assume all to be $\frac{1}{2019}$ \begin{equation} \frac{1}{\frac{20}{2019}}= 101.0 \end{equation} \begin{equation} 100.0 < S < 101.0 \end{equation}

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Assume all to be $\frac{1}{2000}$ \begin{equation} \frac{1}{\frac{22}{2000}} = 90.9 \end{equation} assume all to be $\frac{1}{2021}$ \begin{equation} \frac{1}{\frac{22}{2021}}= 91.9 \end{equation} \begin{equation} 90.9 < S < 91.9 \end{equation}

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Assume all to be $\frac{1}{2000}$ \begin{equation} \frac{1}{\frac{23}{2000}} = 87.0 \end{equation} assume all to be $\frac{1}{2022}$ \begin{equation} \frac{1}{\frac{23}{2022}}= 87.9 \end{equation} \begin{equation} 87.0 < S < 87.9 \end{equation}

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Assume all to be $\frac{1}{2000}$ \begin{equation} \frac{1}{\frac{25}{2000}} = 80.0 \end{equation} assume all to be $\frac{1}{2024}$ \begin{equation} \frac{1}{\frac{25}{2024}}= 81.0 \end{equation} \begin{equation} 80.0 < S < 81.0 \end{equation}

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Assume all to be $\frac{1}{2000}$ \begin{equation} \frac{1}{\frac{26}{2000}} = 76.9 \end{equation} assume all to be $\frac{1}{2025}$ \begin{equation} \frac{1}{\frac{26}{2025}}= 77.9 \end{equation} \begin{equation} 76.9 < S < 77.9 \end{equation}

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Assume all to be $\frac{1}{2000}$ \begin{equation} \frac{1}{\frac{29}{2000}} = 69.0 \end{equation} assume all to be $\frac{1}{2028}$ \begin{equation} \frac{1}{\frac{29}{2028}}= 69.9 \end{equation} \begin{equation} 69.0 < S < 69.9 \end{equation}

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Assume all to be $\frac{1}{2000}$ \begin{equation} \frac{1}{\frac{30}{2000}} = 66.7 \end{equation} assume all to be $\frac{1}{2029}$ \begin{equation} \frac{1}{\frac{30}{2029}}= 67.6 \end{equation} \begin{equation} 66.7 < S < 67.6 \end{equation}

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Assume all to be $\frac{1}{2000}$ \begin{equation} \frac{1}{\frac{33}{2000}} = 60.6 \end{equation} assume all to be $\frac{1}{2032}$ \begin{equation} \frac{1}{\frac{33}{2032}}= 61.6 \end{equation} \begin{equation} 60.6 < S < 61.6 \end{equation}

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Assume all to be $\frac{1}{2000}$ \begin{equation} \frac{1}{\frac{34}{2000}} = 58.8 \end{equation} assume all to be $\frac{1}{2033}$ \begin{equation} \frac{1}{\frac{34}{2033}}= 59.8 \end{equation} \begin{equation} 58.8 < S < 59.8 \end{equation}

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Assume all to be $\frac{1}{2000}$ \begin{equation} \frac{1}{\frac{36}{2000}} = 55.6 \end{equation} assume all to be $\frac{1}{2035}$ \begin{equation} \frac{1}{\frac{36}{2035}}= 56.5 \end{equation} \begin{equation} 55.6 < S < 56.5 \end{equation}