Statement-1: The sum of the series 1+(1+2+4)+(4+6+9)+(9+12+16)+…+(361+380+400) is  8000Statement-2: ∑k=1n k3−(k−1)3=n3 for each natural number n.

# Statement$-1$: The sum of the series $1+\left(1+2+4\right)+\left(4+6+9\right)+\left(9+12+16\right)+\dots +\left(361+380+400\right)$ is Statement$-2$: $\sum _{k=1}^{n} \left({k}^{3}-\left(k-1{\right)}^{3}\right)={n}^{3}$ for each natural number $n.$

1. A

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

2. B

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

3. C

STATEMENT-1 is True, STATEMENT-2 is False

4. D

STATEMENT-1 is False, STATEMENT-2 is True

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### Solution:

Statement$-2$ is true since

Putting $n=20,$ we get

$\sum _{k=1}^{20} \left(\left(k-1{\right)}^{2}+k\left(k-1\right)+{k}^{2}\right)={20}^{3}$

Thus, Statement$-1$ is also true and Statement$-2$ is a correct reason for it.

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