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Alternating fraction sums

Solve the first problem 18 problems

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1. Sum of squares divided by 365 ●○○○○ 10 ptsWhat is the value of the expression $\frac{10^{2}+11^{2}+12^{2}+13^{2}+14^{2}}{365}$?
2. Ratio of alternating sums ●○○○○ 10 ptsFind the value of $\frac{2-4+6-8+10-12+14}{3-6+9-12+15-18+21}$ as a fraction in lowest terms. Give the numerator and the denominator.
3. Ratio of alternating power-of-two sums ●○○○○ 10 ptsCompute $\dfrac{2+4-8+16+32-64}{4+8-16+32+64-128}$. Give your answer as a decimal.
4. Sum of fractions with shared denominators ●●○○○ 20 ptsFind the value of $\left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{20}\right)+\left(\frac{2}{3}+\cdots+\frac{2}{20}\right)+\cdots+\left(\frac{18}{19}+\frac{18}{20}\right)+\frac{19}{20}$.
5. Telescoping sum of odd-numerator fractions ●●○○○ 20 ptsCalculate $\frac{3}{4}+\frac{5}{36}+\frac{7}{144}+\frac{9}{400}+\frac{11}{900}+\frac{13}{1764}+\frac{15}{3136}$. Give your answer as a decimal.
6. Nested fraction sum with halving ●●○○○ 20 ptsDetermine the value of $2002+\frac{1}{2}\left(2001+\frac{1}{2}\left(2000+\cdots+\frac{1}{2}\left(3+\frac{1}{2}\cdot 2\right)\right)\cdots\right)$.
7. Ratio of weighted harmonic sums ●●●○○ 50 ptsDetermine the value of $\frac{\frac{2016}{1} + \frac{2015}{2} + \frac{2014}{3} + \dots + \frac{1}{2016}}{\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{2017}}$.
8. Nested half-fraction sum ●●●○○ 50 ptsFind the value of $2002 + \frac{1}{2} \left( 2001 + \frac{1}{2} \left( 2000 + \dots + \frac{1}{2} \left( 3 + \frac{1}{2} \cdot 2 \right) \right) \dotsb \right)$.
9. Alternating-parity sum of reciprocal squares ●●●○○ 50 ptsCompute the value of $$\frac{1+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\cdots+\frac{1}{997^2}+\frac{1}{999^2}-\frac{1}{1002^2}-\frac{1}{1004^2}-\frac{1}{1006^2}-\cdots-\frac{1}{1998^2}-\frac{1}{2000^2}}{1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots+\frac{1}{999^2}+\frac{1}{1000^2}}$$ without using a calculator. Give your answer as a decimal.

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