Statement-1: The sum of the series 1 + (1 + 2 | vedclass

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(C) For Statement-$2$: The sum is a telescoping series: $\sum_{k=1}^n (k^3 - (k-1)^3) = (1^3 - 0^3) + (2^3 - 1^3) + \dots + (n^3 - (n-1)^3) = n^3$. Th

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Statement-$1$ is true,Statement-$2$ is true,Statement-$2$ is the correct explanation for Statement-$1$.

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(C) For Statement-$2$: The sum is a telescoping series: $\sum_{k=1}^n (k^3 - (k-1)^3) = (1^3 - 0^3) + (2^3 - 1^3) + \dots + (n^3 - (n-1)^3) = n^3$. Thus,Statement-$2$ is true.For Statement-$1$: The $k$-th term of the series (starting from $k=1$) is $T_k = (k-1)^2 + (k-1)k + k^2$. Note that $(k-1)^2 + (k-1)k + k^2 = \frac{k^3 - (k-1)^3}{k - (k-1)} = k^3 - (k-1)^3$.The series is $\sum_{k=1}^{20} (k^3 - (k-1)^3) = 20^3 = 8000$. Since the last term is $(361 + 380 + 400) = 19^2 + 19 	imes 20 + 20^2$,this corresponds to $k=20$. Thus,the sum is $8000$. Statement-$1$ is true and Statement-$2$ is the correct explanation.

Statement-$1$: The sum of the series $1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) + \dots + (361 + 380 + 400)$ is $8000$.
Statement-$2$: For any natural number $n$,$\sum_{k=1}^n (k^3 - (k-1)^3) = n^3$.

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Statement-$1$: The sum of the series $1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) + \dots + (361 + 380 + 400)$ is $8000$.Statement-$2$: For any natural number $n$,$\sum_{k=1}^n (k^3 - (k-1)^3) = n^3$.