Chapter 8.1, Problem 8E

To determine

To find: Formula for the general term $a_n$ of the sequence.

Answer to Problem 8E

The formula for the general term of the sequence is $a_n={\left(-1\right)}^n\frac{n}{{\left(n+1\right)}^2}$ .

Explanation of Solution

Given information: The sequence is $\left\{-\frac{1}{4},\frac{2}{9},-\frac{3}{16},\frac{4}{25},\mathrm{....}\right\}$

Calculation:

Here, $$

  $a_1=-\frac{1}{4},a_2=\frac{2}{9},a_3=-\frac{3}{16},a_4=\frac{4}{25}$

The numerator in first term is 1, in second term is 2, in third term is 3;

So, in general the n^th term will have numerator n.

The denominators are squares of numbers, in first term denominator is 2², in second term denominator is 3², in third term denominator is 4²;

So, in general the n^th term will have denominator ${\left(n+1\right)}^2$ .

The signs of the terms are alternately negative and positive, so the general term will be multiplied by ${\left(-1\right)}^n$ .

  $\therefore$ General term $a_n={\left(-1\right)}^n\frac{n}{{\left(n+1\right)}^2}$

Hence, the formula for the general term of the sequence is $a_n={\left(-1\right)}^n\frac{n}{{\left(n+1\right)}^2}$ .