First slide

Special Series in Sequences and Series

Question

Sum to $n$ terms of the series $\frac{1}{(1 + x) (1 + 2 x)} + \frac{1}{(1 + 2 x) (1 + 3 x)} + \frac{1}{(1 + 3 x) (1 + 4 x)} + \dots$ is

Moderate

A

$\frac{n x}{(1 + x) (1 + n x)}$
B

$\frac{n}{(1 + x) [1 + (n + 1) x]}$
C

$\frac{x}{(1 + x) (1 + (n - 1) x)}$
D

none of these

Solution

If $t_{r}$ denotes the $r t h$ term of the series, then
$x t_{r} = \frac{x}{(1 + r x) (1 + (r + 1) x)} = \frac{1}{1 + r x} - \frac{1}{1 + (r + 1) x}$
$\begin{array}{l} \Rightarrow x \sum_{r = 1}^{n} t_{r} = \sum_{r = 1}^{n} [\frac{1}{1 + r x} - \frac{1}{1 + (r + 1) x}] \\ = \frac{1}{1 + x} - \frac{1}{1 + (n + 1) x} \\ = \frac{n x}{(1 + x) (1 + (n + 1) x)} \end{array}$
$\Rightarrow \sum_{r = 1}^{n} t_{r} = \frac{n}{(1 + x) [1 + (n + 1) x]}$

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