Chapter 11.6, Problem 6WE

To determine

To calculate:

Sum of given geometric series.

Answer to Problem 6WE

Sum of given geometric series is $2500$

Explanation of Solution

Give information:

  $500+400+320+256+\mathrm{...}$

Calculation:

In order to find out common ratio of given number,

  $500+400+320+256+\mathrm{...}$

Here first term is $a_1=500$

So common ratio is $r=\frac{400}{500}=0.8$

Since $\left|r\right|=\left|0.8\right|<1$

Infinite sum exists.

So, Sum of given geometric series is calculated as,

  $=\frac{a^n}{1-r}$

Where,

  $a_1=$ Initial term.

  $r=$ common ratio between terms.

  $n=$ number of terms

Put the values in the equation.

  $\begin{array}{l}=\frac{a^n}{1-r}\\ =\frac{500}{1-\left(0.8\right)}\\ =\frac{500}{0.2}\\ =2500\end{array}$

Therefore, sum of given geometric series is $2500$ .