First slide

Special Series in Sequences and Series

Question

$The sum to n terms of the series \frac{3}{1^{2}} + \frac{5}{1^{2} + 2^{2}} + \frac{7}{1^{2} + 2^{2} + 3^{2}} + \dots is$

Moderate

A

$\frac{6 n}{n + 1}$
B

$\frac{6}{n + 1}$
C

$\frac{6 (n + 1)}{n}$
D

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

Solution

$T_{n} = \frac{(2 n + 1) 6}{n (n + 1) (2 n + 1)} = 6 (\frac{1}{n} - \frac{1}{n + 1}) ∴ S_{n} = \sum T_{n} = \frac{6 n}{n + 1}$

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