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  4. The sum to n terms of [(1/1.3) + (2/1.3 . 5) + (3/1.3 . 5.7) + (4/1.3 . 5.7 . 9) + . . . . . . . .] is

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Q. The sum to $n$ terms of $\left[\frac{1}{1.3} + \frac{2}{1.3 . 5} + \frac{3}{1.3 . 5.7} + \frac{4}{1.3 . 5.7 . 9} + . . . . . . . .\right]$ is

NTA AbhyasNTA Abhyas 2022

Solution:

$t_{n}=\frac{n}{1.3 . 5 . \, . . . . . . . . . \left(2 n + 1\right)}$
$= \, \frac{1}{2}. \, \frac{\left(2 n + 1\right) - 1}{1.3 . 5 . \, . . . . . . . . . \left(2 n + 1\right)}$
$=\frac{1}{2}\left[\frac{1}{1.3 . 5 . . . . . . \left(2 n - 1\right)} - \frac{1}{1.3 . 5 . . . . . . . \left(2 n + 1\right)}\right]$
$=\frac{1}{2}\left(T_{n - 1} - \, T_{n}\right)$
$\therefore 2t_{n}=T_{n - 1}- \, T_{n}\ldots \left(1\right) \, \left(w h e r e \, T_{n} = \frac{1}{1.3 . 5 . . . . . . . \left(2 n + 1\right)}\right)$

$\therefore \, \, 2 S_{n} \, = \, \displaystyle \sum _{n = 2}^{n}2 t_{n} + 2 t_{1}$
$\Rightarrow \, 2 S_{n} \, - \, 2 t_{1} \, = \, \left(\right. T_{1} \, - \, T_{2} \left.\right) + \left(\right. T_{2} \, - \, T_{3} \left.\right) + ........ + \left(\right. T_{n - 1} - T_{n} \left.\right)$
$\Rightarrow \, 2 S_{n} \, - \, 2 t_{1} \, = \, T_{1} - T_{n}$
$\Rightarrow \, 2S_{n} \, = \, 2t_{1} \, + \, T_{1}-T_{n} \\ \, \, \, = \, 2\times \frac{1}{3}+\frac{1}{1 \times 3}-\frac{1}{1 \times 3 \times 5..... \left(\right. 2 n + 1 \left.\right)}$
$\Rightarrow \, S_{n} \, = \, \frac{1}{2}\left(1 - \frac{1}{1 \times 3 \times 5 ....... \left(\right. 2 n + 1 \left.\right)}\right)$