Chapter 9.1, Problem 43E

To determine

To Find :Nth term of the arithmetic sequence

  $1,\frac{1}{4},\frac{1}{9},\frac{1}{16},\frac{1}{25}\mathrm{....}$

Answer to Problem 43E

  $a_n=\frac{1}{n^2}$

Explanation of Solution

Given:Write an expression for the apparent nth term of the sequence

  $1,\frac{1}{4},\frac{1}{9},\frac{1}{16},\frac{1}{25}\mathrm{....}$

  $\begin{array}{l}1,\frac{1}{4},\frac{1}{9},\frac{1}{16},\frac{1}{25}\mathrm{....}\\ \end{array}$

The series form a pattern as

  $\begin{array}{l}1,\frac{1}{4},\frac{1}{9},\frac{1}{16},\frac{1}{25}\mathrm{....}\\ \frac{1}{1^2},\frac{1}{2^2},\frac{1}{3^2},\frac{1}{4^2},\frac{1}{5^2}\\ therefore,\\ a_n=\frac{1}{n^2}\end{array}$

Hence , The nth term of the series is $a_n=\frac{1}{n^2}$