Sequences and series › Geometric sequences

Common ratio

Common ratio from an arithmetic condition ●○○○○ 10 ptsFor a geometric sequence $\{a_n\}$ with sum of the first $n$ terms $S_n$ , we have $a_1=1$ , and $a_1$ , $S_2$ , $5$ (in this order) form an arithmetic sequence. Find the common ratio $q$ .
Common ratio of a geometric sequence ●○○○○ 10 ptsIn a geometric sequence $\{a_n\}$ where all terms are positive, $a_6 = a_5 + 2a_4$ . Find the common ratio $q$ .
Common ratio from two terms ●○○○○ 10 ptsIn the geometric sequence $\{a_n\}$ , $a_3=2$ and $a_6=\frac{1}{4}$ . Find the common ratio of the sequence. Give your answer as a decimal.
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 $\frac{a_3}{a_1}$ .
Ratio of sums in a geometric sequence ●●○○○ 20 ptsA geometric sequence $\{a_n\}$ has all positive terms and satisfies $a_3 = a_1 + 2a_2$ . Find the value of $\frac{a_9+a_{10}}{a_7+a_8}$ . Round to two decimal places.
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.
Unique geometric sequence with shifted terms ●●●○○ 50 ptsThe geometric sequence $\{a_n\}$ satisfies $a_1=a$ with $a>0$ , and $a_1+1$ , $a_2+2$ , $a_3+3$ form a geometric sequence. If $\{a_n\}$ is unique, find the value of $a$ . Round to two decimal places.
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.
Ratio in a geometric sequence ●●●○○ 50 ptsIn the geometric sequence $\{a_n\}$ of real numbers, $a_5 \cdot a_{11} = 4$ and $a_3 + a_{13} = 5$ . Find all possible values of $\frac{a_{14}}{a_4}$ . Give each value as a decimal, in increasing order.

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