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(D) The general term of the given series is $T_r = (-1)^r (3 + 5r) { }^n C_r$ for $r = 0, 1, 2, \ldots, n$. The sum $S_n$ is given by $S_n = \sum_{r=0

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(D) The general term of the given series is $T_r = (-1)^r (3 + 5r) { }^n C_r$ for $r = 0, 1, 2, \ldots, n$. The sum $S_n$ is given by $S_n = \sum_{r=0}^n (-1)^r (3 + 5r) { }^n C_r$. $S_n = 3 \sum_{r=0}^n (-1)^r { }^n C_r + 5 \sum_{r=0}^n (-1)^r r { }^n C_r$. For $n > 1$,the first part $3 \sum_{r=0}^n (-1)^r { }^n C_r = 3(1 - 1)^n = 0$. For the second part,using $r { }^n C_r = n { }^{n-1} C_{r-1}$,we get $5n \sum_{r=1}^n (-1)^r { }^{n-1} C_{r-1} = 5n \sum_{k=0}^{n-1} (-1)^{k+1} { }^{n-1} C_k = -5n (1 - 1)^{n-1} = 0$. Thus,$S_n = 0 + 0 = 0$.

If $n$ is a positive integer greater than $1$,then $3({ }^n C_0) - 8({ }^n C_1) + 13({ }^n C_2) - 18({ }^n C_3) + \ldots$ up to $(n+1)$ terms $=$

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