Chapter 12.1, Problem 8E

To determine

Calculate the first four and the $100th$ term of the given sequence.

Answer to Problem 8E

The first four term of the sequence are $1,\frac{1}{4},\frac{1}{9},\frac{1}{16}$ and the $100th$ term are $\frac{1}{10000}$ .

Explanation of Solution

Given:

It is given in the question that the sequence is $a_n=\frac{1}{n^2}$ .

Concept Used:

In this, use the concept of arithmetic progression in which put the number and calculate the problem.

Calculation:

Here, the sequence is $a_n=\frac{1}{n^2}$ .

To find the first four terms, substitute $n=1,2,3and4$ in the formula for the nth term

  $a_n=\frac{1}{n^2}$ ( nth term)

  $\begin{array}{l}{a}_1=\frac{1}{{(1)}^2}=1(n=1)\\ a_2=\frac{1}{{(2)}^2}=\frac{1}{4}(n=2)\\ a_3=\frac{1}{{(3)}^2}=\frac{1}{9}(n=3)\\ a_4=\frac{1}{{(4)}^2}=\frac{1}{16}(n=4)\end{array}$

Now, to find the $100th$ term, substitute $n=100$ ,

  $a_{100}=\frac{1}{{(100)}^2}=\frac{1}{10000}$

Conclusion:

The first four term of the sequence are $1,\frac{1}{4},\frac{1}{9},\frac{1}{16}$ and the $100th$ term are $\frac{1}{10000}$ .