The sum of the series 3×12+5×22+7×32+… is

# The sum of the series $3×{1}^{2}+5×{2}^{2}+7×{3}^{2}+\dots$ is

1. A

$\frac{\mathrm{n}\left(\mathrm{n}+1\right)\left({\mathrm{n}}^{2}-5\mathrm{n}-1\right)}{6}$

2. B

$\frac{\mathrm{n}\left(\mathrm{n}+1\right)\left(3{\mathrm{n}}^{2}+5\mathrm{n}+1\right)}{6}$

3. C

$\frac{\mathrm{n}\left(\mathrm{n}-1\right)\left(3{\mathrm{n}}^{2}-5\mathrm{n}-1\right)}{6}$

4. D

None of these

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### Solution:

Let given series is 5 = 3 x 12 + 5 x 22 + 7 x 32 + ...
First, we will split the given series into two parts which are 3, 5,7, ... and 12,22,32, , . . and find the nth term of each part
separately to find the nth term of the given series.

Tn = (rth term of 3,5, 7,. . ) x (nth term of 1,2,3, . . )2

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