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

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

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

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

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

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

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.

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

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.