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Q.
The sum of the series $\frac{1}{1.3 .5}+\frac{1}{3.5 .7}+\frac{1}{5.7 .9}+\ldots .$. to $n$ terms is
Sequences and Series
Solution:
$S =\frac{1}{1.3 .5}+\frac{1}{3.5 .7}+\frac{1}{5.7 .9}+\ldots . . n$
$T _{ n }=\frac{1}{(2 n -1)(2 n +1)(2 n +3)}$
$\therefore T _{ n }=\frac{1}{4}\left[\frac{1}{(2 n -1)(2 n +1)}-\frac{1}{(2 n +1)(2 n +3)}\right]$
$T _1=\frac{1}{4}\left[\frac{1}{1.3}-\frac{1}{3.5}\right], T _2=\frac{1}{4}\left[\frac{1}{3.5}-\frac{1}{5.7}\right]$
$T _3=\left[\frac{1}{5.7}-\frac{1}{7.9}\right]$
$T _{ n }=\left[\frac{1}{(2 n-1)(2 n+1)}-\frac{1}{(2 n+1)(2 n+3)}\right]$
sum of all terms gives $S _{ n }$
$S _{ n }=\frac{1}{4}\left[\frac{1}{3}-\frac{1}{(2 n+1)(2 n+3)}\right]$