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

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(D) Statement $-2$: $\sum_{k=1}^{n} (k^3 - (k-1)^3)$ is a telescoping sum. Expanding it: $(1^3 - 0^3) + (2^3 - 1^3) + (3^3 - 2^3) + \dots + (n^3 - (n-

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

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(D) Statement $-2$: $\sum_{k=1}^{n} (k^3 - (k-1)^3)$ is a telescoping sum. Expanding it: $(1^3 - 0^3) + (2^3 - 1^3) + (3^3 - 2^3) + \dots + (n^3 - (n-1)^3) = n^3$. Thus,Statement $-2$ is true.Statement $-1$: The series is $\sum_{k=1}^{20} ((k-1)^2 + (k-1)k + k^2)$.Note that $(k-1)^2 + (k-1)k + k^2 = \frac{k^3 - (k-1)^3}{k - (k-1)} = k^3 - (k-1)^3$.For $k=1$,$T_1 = 1^3 - 0^3 = 1$.For $k=2$,$T_2 = 2^3 - 1^3 = 7 = 1+2+4$.For $k=20$,$T_{20} = 20^3 - 19^3 = 8000 - 6859 = 1141 = 361 + 380 + 400$.The sum is $\sum_{k=1}^{20} (k^3 - (k-1)^3) = 20^3 = 8000$. Thus,Statement $-1$ is true.Since Statement $-1$ is derived directly from the identity in Statement $-2$,Statement $-2$ is the correct explanation for Statement $-1$.

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

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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 $-2$: $\sum_{k=1}^{n} (k^3 - (k-1)^3) = n^3$,for any natural number $n$.