Chapter 8.3, Problem 65E

To determine

To write: the sum $5+15+45+\mathrm{...}+3645$ by using summation notation.

Answer to Problem 65E

The summation notation for the sum $5+15+45+\mathrm{...}+3645$ is $5{\displaystyle \sum _{n=1}^7{3}^{n-1}}$ .

Explanation of Solution

Given information :

Givensum is $5+15+45+\mathrm{...}+3645$

As the given series is $5+15+45+\mathrm{...}+3645$ .

The above series can be simplified as

  $\begin{array}{c}5+15+45+\mathrm{...}+3645=5\left(1+3+9+\mathrm{.......}+729\right)\\ =5\left(1+3+3^2+\mathrm{.......}+3^6\right)\\ =5{\displaystyle \sum _{n=1}^7{3}^{n-1}}\end{array}$

Hence,the summation notation of the sum $5+15+45+\mathrm{...}+3645$ is $5{\displaystyle \sum _{n=1}^7{3}^{n-1}}$ .