SATHEE: Sequences and Series 2 Question 15

Sequences and Series 2 Question 15

16. The sum of the first $n$ terms of the series $1^{2} + 2 \cdot 2^{2} + 3^{2} + 2 \cdot 4^{2} + 5^{2} + 2 \cdot 6^{2} + \dots$ is $\frac{n (n + 1)^{2}}{2}$ , when $n$ is even. When $n$ is odd, the sum is …. .

(1988, 2M)

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Answer:

Correct Answer: 16. (9)

Solution:

Here, $1^{2} + 2 \cdot 2^{2} + 3^{2} + 2 \cdot 4^{2} + 5^{2} + \dots$ upto $n$ terms

$= \frac{n (n + 1)^{2}}{2}$

[when $n$ is even] … (i)

When $n$ is odd, $1^{2} + 2 \cdot 2^{2} + 3^{2} + 2 \cdot 4^{2} + 5^{2} \dots + n^{2}$

$\begin{aligned} = 1^{2} + 2 \cdot 2^{2} + 3^{2} + 2 \cdot 4^{2} + \dots + 2 (n - 1)^{2} + n^{2} \\ = \frac{(n - 1) (n)^{2}}{2} + n^{2} [from Eq. (i)] \\ = n^{2} \frac{n - 1}{2} + 1 = n^{2} \frac{(n + 1)}{2} \end{aligned}$

$∴ 1^{2} + 2 \cdot 2^{2} + 3^{2} + 2 \cdot 4^{2} + \dots$ upto $n$ terms, when $n$ is odd

$= \frac{n^{2} (n + 1)}{2}$

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Sequences and Series 2 Question 15

16. The sum of the first n terms of the series 12+2⋅22+32+2⋅42+52+2⋅62+… is n(n+1)22, when n is even. When n is odd, the sum is …. .

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16. The sum of the first $n$ terms of the series $1^{2} + 2 \cdot 2^{2} + 3^{2} + 2 \cdot 4^{2} + 5^{2} + 2 \cdot 6^{2} + \dots$ is $\frac{n (n + 1)^{2}}{2}$ , when $n$ is even. When $n$ is odd, the sum is …. .