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1. Difference of a linear function ●○○○○ 10 ptsThe function $f$ is linear and satisfies $f(d+1)-f(d) = 3$ for all real numbers $d$. What is $f(3)-f(5)$?
2. Sum after repeated difference insertion ●●○○○ 20 ptsWrite the 4 numbers $1,9,8,8$ in a row and perform the following operation: for each pair of adjacent numbers, subtract the left number from the right number, and write the result between the two numbers. This completes one operation. Then perform the same operation on the 7 numbers now in a row, and continue this process for a total of 100 operations. Find the sum of the final row of numbers.
3. 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.
4. Nested fraction sum with halving ●●○○○ 20 ptsDetermine the value of $2002+\frac{1}{2}\left(2001+\frac{1}{2}\left(2000+\cdots+\frac{1}{2}\left(3+\frac{1}{2}\cdot 2\right)\right)\cdots\right)$.
5. Ratio of weighted harmonic sums ●●●○○ 50 ptsDetermine the value of $\frac{\frac{2016}{1} + \frac{2015}{2} + \frac{2014}{3} + \dots + \frac{1}{2016}}{\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{2017}}$.
6. Nested half-fraction sum ●●●○○ 50 ptsFind the value of $2002 + \frac{1}{2} \left( 2001 + \frac{1}{2} \left( 2000 + \dots + \frac{1}{2} \left( 3 + \frac{1}{2} \cdot 2 \right) \right) \dotsb \right)$.
7. 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.