Chapter 11.6, Problem 13WE

To determine

To calculate:

Value of $S_1,S_2,S_3,S_4,S_5$ and $S$ .

Answer to Problem 13WE

Value of $S$ is $\frac{32}{8}$ and $S_1=8,S_2=10,S_3=10\frac{1}{2},S_4=\frac{85}{8},S_5=\frac{341}{32}$

Explanation of Solution

Give information:

  $8+2+\frac{1}{2}+\frac{1}{8}+\mathrm{...}$

Calculation:

Given is $8+2+\frac{1}{2}+\frac{1}{8}+\mathrm{...}$ ,

  $\begin{array}{l}{S}_1=8\\ S_2=8+2\\ =10\\ S_3=10+\frac{1}{2}\\ =10\frac{1}{2}\\ S_4=10\frac{1}{2}+\frac{1}{8}\\ S_4=\frac{85}{8}\\ S_5=\frac{85}{8}+\frac{1}{32}\\ S_5=\frac{341}{32}\end{array}$

The formula for sum of an infinite geometric series is,

  $S=\frac{s_1}{1-r}$

Where,

  $r=$ common ratio.

  $s_1=$ first term.

Since $-1<r<1$

Here common ratio is $=\frac{1}{4}$

Then put the value in equation of $S$ ,

  $\begin{array}{l}S=\frac{8}{1-\left(\frac{1}{4}\right)}\\ S=\frac{8}{\left(\frac{3}{4}\right)}\\ S=\frac{32}{8}\end{array}$

Therefore, Value of $S$ is $\frac{32}{8}$ .