Chapter 11, Problem 39SGR

To determine

Find the sum of the series $243+81+27+\mathrm{.....}$ to 5 terms

Answer to Problem 39SGR

  ${\text{S}}_5=363$

Explanation of Solution

Given:

Find the sum of the series $243+81+27+\mathrm{.....}$ to 5 terms

Concept Used:

The series: $243+81+27+\mathrm{.....}$

Find the sum of the geometric series for which $a_1=243,r=\frac{81}{243}=\frac{1}{3},\text{and}n=5$

The formula for sum of nth term is ${\text{S}}_n=\frac{a_1(1-r^n)}{1-r}$

Calculation:

  $a_1=243,r=\frac{1}{3},n=5;\text{find}{\text{S}}_5$

  $\begin{array}{l}{\text{S}}_n=\frac{a_1(1-r^n)}{1-r}\\ {\text{S}}_5=\frac{243(1-{(\frac{1}{3})}^5)}{1-(\frac{1}{3})}=\frac{243·(\frac{242}{243})}{\frac{2}{3}}=363\end{array}$  ${\text{S}}_5=363$

Thus, Sum of 5 terms of the series $243+81+27+\mathrm{.....}$ is 363