# How do you find general term formula for {-1/4, 2/9, -3/16, 4/25, ...}?

Mar 24, 2016

${n}^{t h}$ term of the series $\left\{- \frac{1}{4} , \frac{2}{9} , - \frac{3}{16} , \frac{4}{25} , \ldots\right\}$ is$\frac{{\left(- 1\right)}^{n} \times n}{n + 1} ^ 2$

#### Explanation:

Here one desires is the ${n}^{t h}$ term of the series $\left\{- \frac{1}{4} , \frac{2}{9} , - \frac{3}{16} , \frac{4}{25} , \ldots\right\}$.

As is observed the numerators are $\left\{- 1 , 2 , - 3 , 4 , \ldots\right\}$. Hence numerator of ${n}^{t h}$ term is $n$ if $n$ is odd and $- n$ if $n$ is even.

Hence we can write numerator of ${n}^{t h}$ term as ${\left(- 1\right)}^{n} \times n$.

The denominators are $\left\{4 , 9 , 16 , 25 , \ldots\right\}$ i.e. ${n}^{t h}$ term is the square of $\left(n + 1\right)$ or ${\left(n + 1\right)}^{2}$.

Hence general formula for ${n}^{t h}$ term of the series $\left\{- \frac{1}{4} , \frac{2}{9} , - \frac{3}{16} , \frac{4}{25} , \ldots\right\}$ is $\frac{{\left(- 1\right)}^{n} \times n}{n + 1} ^ 2$