Sequences and Series 2 Question 15
16. The sum of the first $n$ terms of the series $1^{2}+2 \cdot 2^{2}+3^{2}+2 \cdot 4^{2}+5^{2}+2 \cdot 6^{2}+\ldots$ 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, $1^{2}+2 \cdot 2^{2}+3^{2}+2 \cdot 4^{2}+5^{2}+\ldots$ upto $n$ terms
$$ =\frac{n(n+1)^{2}}{2} $$
[when $n$ is even] … (i)
When $n$ is odd, $1^{2}+2 \cdot 2^{2}+3^{2}+2 \cdot 4^{2}+5^{2} \ldots+n^{2}$
$$ \begin{aligned} & ={1^{2}+2 \cdot 2^{2}+3^{2}+2 \cdot 4^{2}+\ldots+2(n-1)^{2} }+n^{2} \\ & =\frac{(n-1)(n)^{2}}{2}+n^{2} \quad \text { [from Eq. (i)] } \\ & =n^{2} \frac{n-1}{2}+1=n^{2} \frac{(n+1)}{2} \end{aligned} $$
$\therefore 1^{2}+2 \cdot 2^{2}+3^{2}+2 \cdot 4^{2}+\ldots$ upto $n$ terms, when $n$ is odd
$$ =\frac{n^{2}(n+1)}{2} $$
