Question Type 2: Given the sum to infinity for a sequence, finding the initial value or common ratio Exercises

The sum to infinity of a geometric progression is given by $S(r)=\frac{3}{1-r}-2$ for $|r|<1$. Find the first term $a$ in terms of $r$.

A geometric sequence has first term $r^2$ and common ratio $r$, and its sum to infinity equals $r$. Find all possible values of $r$.

The sum to infinity of a geometric series is expressed as $S = \frac{2k-1}{k}$, the first term is $4$ and the common ratio is $k$. Find $k$.

For a geometric progression with first term $1$ and common ratio $r$, the sum to infinity is equal to the sum of the first four terms. Find the value of $r$.

The sum to infinity of a geometric sequence is $12$ and its first term is $3$. Find the common ratio $r$.

A geometric sequence has first term $8$ and positive common ratio $r$. The sum to infinity of its terms starting from the third term is $16$. Find $r$.

For a geometric sequence, the sum of the first three terms is $21$ and the sum to infinity is $42$. Find the first term $u_1$ and the common ratio $r$.

The sum to infinity of a geometric series is $25$ and its second term is $5$. Find the first term and the common ratio.

The sum to infinity of a geometric sequence is $9$ and the sum of its first two terms is $6$. Find the first term and common ratio.

Find the common ratio.

Given a geometric series with sum to infinity $31.25$ and its fifth term $2.56$, find the first term $a$ and the common ratio $r$.

The sum to infinity of a geometric series is $20$ and the common ratio is $0.8$. Find its first term.