Statement-1: The sum of the series 1+(1+2+4)+ | vedclass

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

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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-$2$: The sum is a telescoping series: $\sum_{k=1}^{n} (k^3 - (k-1)^3) = (1^3 - 0^3) + (2^3 - 1^3) + \dots + (n^3 - (n-1)^3) = n^3$. This is true.Statement-$1$: The $k$-th term of the series is $T_k = (k-1)^2 + (k-1)k + k^2 = k^2 - 2k + 1 + k^2 - k + k^2 = 3k^2 - 3k + 1$.We know that $k^3 - (k-1)^3 = 3k^2 - 3k + 1$. Thus,$T_k = k^3 - (k-1)^3$.The series is $\sum_{k=1}^{20} T_k = \sum_{k=1}^{20} (k^3 - (k-1)^3) = 20^3 = 8000$.Since the last term is $361+380+400 = 19^2 + 19 	imes 20 + 20^2$,this corresponds to $k=20$.Both statements are true and Statement-$2$ explains 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 $n$.