Question

What is the sum of an infinite geometric series if the first term is 81 and the common ratio is 2⁄3?

Answer 1

The terms in the series would be

$81+81\left(\dfrac23\right)+81\left(\dfrac23\right)^2+81\left(\dfrac23\right)^3+\cdots$

Consider the first $n$ terms of this series, and call it

$S_n=81+81\left(\dfrac23\right)+81\left(\dfrac23\right)^2+\cdots+81\left(\dfrac23\right)^{n-1}$

We have

$\dfrac23S_n=81\left(\dfrac23\right)+81\left(\dfrac23\right)^2+81\left(\dfrac23\right)^3+\cdots+81\left(\dfrac23\right)^n$

and subtracting this from $S_n$ gives

$S_n-\dfrac23S_n=81-81\left(\dfrac23\right)^n$

$\dfrac13S_n=81\left(1-\left(\dfrac23\right)^n\right)$

$S_n=243\left(1-\left(\dfrac23\right)^n\right)$

As $n\to\infty$, the $\left(\dfrac23\right)^n$ term will converge to 0 and leave you with

$\displaystyle\lim_{n\to\infty}S_n=243$