Question Type 1: Finding the sum of infinite convergent geometric sequences Exercises

The sum to infinity of a geometric series is $20$ and its first term is $5$. Find the common ratio $r$, and state why the series converges.

Find the sum to infinity of the geometric sequence with first term $u_1 = 5$ and common ratio $r = \frac{1}{3}$. State the condition for convergence.

Find the sum to infinity of the sequence defined by $u_n=7\,(0.2)^{n-1}$.

Calculate the sum to infinity of the series $10+5+2.5+\dots$.

The common ratio $r$ satisfies $2r=1-r^2$ and $|r|<1$. If $u_1=5$, find the sum to infinity of the corresponding geometric series, giving your answer in exact form with a rational denominator.

Given the first three terms of a geometric sequence are $12, \ 4, \ \frac{4}{3}$, find its common ratio and the sum to infinity.

A geometric series has common ratio $r=\frac{2}{5}$ and sum to infinity $15$. Find its first term.

Express the repeating decimal $0.\overline{36}$ as a fraction by interpreting it as an infinite geometric series.

A geometric sequence has terms $6, -3, 1.5,\dots$. Determine its sum to infinity.

A geometric series is given by $4+4k+4k^2+\cdots$ and its sum to infinity is $12$. Determine $k$.

Determine whether the infinite series with first term $u_1 = 8$ and common ratio $r = -\frac{1}{4}$ converges, and if so, find its sum.

Find the sum to infinity of the geometric series with $u_1 = 3$ and common ratio $r = 0.75$.