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(\frac23\right)+81\left(\frac23\right)^2+81\left(\frac23\right)^3+\cdots`

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

`S_n=81+81\left(\frac23\right)+81\left(\frac23\right)^2+\cdots+81\left(\frac23\right)^(n-1)`

We have

`\frac23S_n=81\left(\frac23\right)+81\left(\frac23\right)^2+81\left(\frac23\right)^3+\cdots+81\left(\frac23\right)^n`

and subtracting this from
`S_n` gives

`S_n-\frac23S_n=81-81\left(\frac23\right)^n`

`\frac13S_n=81\left(1-\left(\frac23\right)^n\right)`

`S_n=243\left(1-\left(\frac23\right)^n\right)`

As
`n\to\infty`, the
`\left(\frac23\right)^n` term will converge to 0 and leave you with

`\displaystyle\lim_(n\to\infty)S_n=243`