1-1-3-2-1-3-5-3-1-3-5-7-n-terms-Find-sum-

Question Number 105250 by bshahid010@gmail.com last updated on 27/Jul/20

\frac{1}{1 \times 3} + \frac{2}{1 \times 3 \times 5} + \frac{3}{1 \times 3 \times 5 \times 7} + \dots \dots \dots n - terms

Find sum

Answered by nimnim last updated on 27/Jul/20

L e t S_{n} = \frac{1}{1 \times 3} + \frac{2}{1 \times 3 \times 5} + \frac{3}{1 \times 3 \times 5 \times 7} + \dots . t o n t e r m s

N r = n^{t h} t e r m o f 1, 2, 3, \dots . . = n

D r = p r o d u c t o f t h e f i r s t (n + 1) t e r m s o f

A P 1, 3, 5 \dots . (2 n + 1), n \in N

∴ t_{n} = \frac{n}{1.3.5 \dots \dots . (2 n + 1)} = \frac{1}{2} [\frac{(2 n + 1) - 1}{1.3.5 \dots . (2 n - 1) (2 n + 1)}]

= \frac{1}{2} [\frac{1}{1.3.5 \dots . . (2 n - 1)} - \frac{1}{1.3.5 \dots . . (2 n + 1)}]

t h e n

t_{1} = \frac{1}{2} [\frac{1}{1} - \frac{1}{1.3}]

t_{2} = \frac{1}{2} [\frac{1}{1.3} - \frac{1}{1.3.5}]

t_{3} = \frac{1}{2} [\frac{1}{1.3.5} - \frac{1}{1.3.5.7}]

\dots \dots \dots \dots . \dots \dots .

\dots \dots \dots \dots . \dots \dots .

t_{n} = \frac{1}{2} [\frac{1}{1.3.5 \dots . (2 n - 1)} - \frac{1}{1.3.5 \dots . (2 n + 1)}]

∴ S_{n} = \frac{1}{2} [1 - \frac{1}{1.3.5 \dots . (2 n + 1)}]

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