Chapter 12.5, Problem 24E

To determine

To write given expression in expanded form and them find its sum.

Answer to Problem 24E

Sum is $16.$

Explanation of Solution

Given information:

  ${\displaystyle \sum _{p=0}^{\infty }4{\left(0.75\right)}^p}$

  $\begin{array}{c}{\displaystyle \sum _{p=0}^{\infty }4{\left(0.75\right)}^p}=4{\left(0.75\right)}^0+4{\left(0.75\right)}^1+4{\left(0.75\right)}^2+\mathrm{...}\\ =4+3+2.25+\mathrm{...}\end{array}$

Series is in geometric progression. Sum of infinite gp is $\frac{a}{1-r}$

  $S=\frac{4}{1-0.75}=\frac{4}{0.25}=16$

Hence the sum of the given series is $16$ .