First slide

Sequence and Series

Question

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

Moderate

A

$\frac{n (n + 1) (n^{2} - 5 n - 1)}{6}$
B

$\frac{n (n + 1) (3 n^{2} + 5 n + 1)}{6}$
C

$\frac{n (n - 1) (3 n^{2} - 5 n - 1)}{6}$
D

None of these

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.
T_n = (rth term of 3,5, 7,. . ) x (nth term of 1,2,3, . . )²
$\begin{array}{l} = [3 + (n - 1) 2] [1 + (n - 1)]^{2} \\ = (3 + 2 n - 2) (n)^{2} = (2 n + 1) n^{2} = 2 n^{3} + n^{2} \\ Now, S = {ΣT}_{n} = 2 {Σn}^{3} + {Σn}^{2} \\ = 2 {[\frac{n (n + 1)}{2}]}^{2} + \frac{n (n + 1) (2 n + 1)}{6} [∵ {Σn}^{3} = {\{\frac{n (n + 1)}{2}\}}^{2}] \\ = \frac{n (n + 1)}{2} [2 \times \frac{n (n + 1)}{2} + \frac{2 n + 1}{3}] \\ = \frac{n (n + 1)}{2} [\frac{3 n (n + 1) + 2 n + 1}{3}] \\ = \frac{n (n + 1)}{6} \times (3 n^{2} + 3 n + 2 n + 1) \\ = \frac{n (n + 1) (3 n^{2} + 5 n + 1)}{6} \end{array}$

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