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6 problems match “geometric series”. Clear

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1. Nested fraction sum with halving ●●○○○ 20 ptsDetermine the value of $2002+\frac{1}{2}\left(2001+\frac{1}{2}\left(2000+\cdots+\frac{1}{2}\left(3+\frac{1}{2}\cdot 2\right)\right)\cdots\right)$.
2. Arithmetic-geometric series in complex z ●●○○○ 20 ptsLet $z=1-2i$. Find the real part and the imaginary part of $\frac{1}{z}+\frac{2}{z^{2}}+\frac{3}{z^{3}}+\cdots$.
3. Ratio of terms in geometric sequence ●●○○○ 20 ptsLet the sum of the first $n$ terms of a geometric sequence $\{a_n\}$ be $S_n$. If $S_8 = 2S_4$, find $\frac{a_3}{a_1}$.
4. Ratio of geometric partial sums ●●○○○ 20 ptsLet $S_{n}$ be the sum of the first $n$ terms of a geometric sequence $\{a_{n}\}$. If $a_{5}+2a_{10}=0$, find the value of $\frac{S_{20}}{S_{10}}$. Give your answer as a decimal.
5. Nested half-fraction sum ●●●○○ 50 ptsFind the value of $2002 + \frac{1}{2} \left( 2001 + \frac{1}{2} \left( 2000 + \dots + \frac{1}{2} \left( 3 + \frac{1}{2} \cdot 2 \right) \right) \dotsb \right)$.
6. Common ratio from a sum relation ●●●○○ 50 ptsLet $S_n$ be the sum of the first $n$ terms of a geometric sequence $\{a_n\}$ with nonzero first term and common ratio $q$. Given that $S_4 = 5S_2$, find all possible values of $q$, listed in increasing order.