First slide

Special Series in Sequences and Series

Question

Find the sum to n terms of the series $1 / (1 \times 3) + 1 / (3 \times 5) + 1 (5 \times 7) + .....$

Moderate

A

$\frac{n}{n + 1}$
B

$\frac{n}{2 n + 1}$
C

$\frac{n}{3 n + 1}$
D

$\frac{n}{5 n + 1}$

Solution

Let $T$ be the rth term of the given series. Then,
$\begin{array}{l} T_{r} = \frac{1}{(2 r - 1) (2 r + 1)}, r = 1, 2, 3 \dots n \\ = \frac{1}{2} (\frac{1}{2 r - 1} - \frac{1}{2 r + 1}) \\ Hence, the required sum is \\ \sum_{i = 1}^{n} T_{r} = \frac{1}{2} [(\frac{1}{1} - \frac{1}{3}) + (\frac{1}{3} - \frac{1}{5}) + (\frac{1}{5} - \frac{1}{7}) + \dots . . . + (\frac{1}{2 n - 1} - \frac{1}{2 n + 1})] \\ = \frac{1}{2} [1 - \frac{1}{2 n + 1}] = \frac{n}{2 n + 1} \end{array}$

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