Assertion (A): 1+(1+2+4)+(4+6+9)+(9+12+16)+ld | vedclass

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(A) The given series is $S = 1 + (1+2+4) + (4+6+9) + (9+12+16) + \ldots + (81+90+100)$. This can be written as: $S = 1 + (1^2 + 1 	imes 2 + 2^2) + (2

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Both $(A)$ and $(R)$ are true and $(R)$ is the correct explanation of $(A)$

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(A) The given series is $S = 1 + (1+2+4) + (4+6+9) + (9+12+16) + \ldots + (81+90+100)$. This can be written as: $S = 1 + (1^2 + 1 	imes 2 + 2^2) + (2^2 + 2 	imes 3 + 3^2) + \ldots + (9^2 + 9 	imes 10 + 10^2)$. The general term of the series for $r=1$ to $10$ is $T_r = (r-1)^2 + r(r-1) + r^2$. Since $r^3 - (r-1)^3 = (r - (r-1))(r^2 + r(r-1) + (r-1)^2) = 1 	imes (r^2 + r(r-1) + (r-1)^2)$,we have $T_r = r^3 - (r-1)^3$. Thus,$S = \sum_{r=1}^{10} (r^3 - (r-1)^3)$. This is a telescoping sum: $(1^3 - 0^3) + (2^3 - 1^3) + (3^3 - 2^3) + \ldots + (10^3 - 9^3) = 10^3 - 0^3 = 1000$. The Reason $(R)$ states $\sum_{r=1}^n (r^3 - (r-1)^3) = n^3$,which is true by the same telescoping property. Therefore,both $(A)$ and $(R)$ are true and $(R)$ is the correct explanation of $(A)$.

Assertion $(A)$: $1+(1+2+4)+(4+6+9)+(9+12+16)+\ldots+(81+90+100)=1000$
Reason $(R)$: $\sum_{r=1}^n(r^3-(r-1)^3)=n^3$ for any natural number $n$.

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Assertion $(A)$: $1+(1+2+4)+(4+6+9)+(9+12+16)+\ldots+(81+90+100)=1000$ Reason $(R)$: $\sum_{r=1}^n(r^3-(r-1)^3)=n^3$ for any natural number $n$.