Chapter 11, Problem 9CT

To determine

To calculate: The sum of the infinite geometric series if it has one:

  $250+150+90+54+\cdot \cdot \cdot$

Answer to Problem 9CT

The sum of the infinite geometric series is $625$

Explanation of Solution

Given information:

  $250+150+90+54+\cdot \cdot \cdot$

First term: $a_1=250$

Formula used:

  $S_{\infty }=\frac{a_1}{1-r}$

Calculation:

Consider the series:

  $250+150+90+54+\cdot \cdot \cdot$

The common ratio is:

  $\begin{array}{l}r=\frac{a_2}{a_1}=\frac{a_3}{a_2}=\frac{a_4}{a_3}\\ r=\frac{150}{250}=\frac{90}{150}=\frac{54}{90}\\ r=0.6\end{array}$

Use the formula of sum of geometric series:

  $S_{\infty }=\frac{a_1}{1-r}$  ... (1)

Put the values of $a_1$ , $r$ and $n$ in equation (1)

  $\begin{array}{l}{S}_{\infty }=\frac{a_1}{1-r}\\ S_{\infty }=\frac{250}{1-0.6}\\ S_{\infty }=\frac{250}{0.4}\\ S_{\infty }=625\end{array}$

The solution of the sum of geometric series is $625$ .