Chapter 12.1, Problem 29E

To determine

Calculate the nth term of a sequence.

Answer to Problem 29E

The nth term of the sequence is $a_n=\frac{2n-1}{n^2}$ .

Explanation of Solution

Given:

It is given in the question the first few terms are $1,\frac{3}{4},\frac{5}{9},\frac{7}{16},\frac{9}{25}\mathrm{........}$ .

Concept Used:

In this, use the concept of understanding the pattern of the first few terms and make the nth sequence.

Calculation:

Here, the first few terms are $1,\frac{3}{4},\frac{5}{9},\frac{7}{16},\frac{9}{25}\mathrm{........}$ .

It is seen that, Denominator of every term of sequence is perfect square and numerator of every term is one less than two times of number of term.

Numerator = $2n-1$ ; Denominator = $n^2$

Common ratio of the sequence : $r=\frac{a_n}{a_{n-1}}=-\frac{25}{5}=-5$

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

Conclusion:

The sequence is $a_n=\frac{2n-1}{n^2}$ .