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If $n$ is a positive integer greater than 1 , then $3\left({ }^n C_0\right)-8\left({ }^n C_1\right)+13\left({ }^n C_2\right)-18\left({ }^n C_3\right)+\ldots$ upto $(n+1)$ terms $=$
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The general term of the given series is
$$
\begin{aligned}
& T_r=(-1)^r(3+5 r)^n C_r, r=0,1,2, \ldots, n \\
& \begin{aligned}
\therefore \quad S_n & =\sum_{r=0}^n(-1)^r(3+5 r)^n C_r \\
& =3 \sum_{r=0}^n(-1)^r \cdot{ }^n C_r+5 \sum_{r=0}^n(-1)^r r^n C_r \\
& =3\left[\sum_{r=0}^n(-1)^r \cdot{ }^n C_r\right]+5\left[\sum_{r=1}^n(-1)^r \cdot r \frac{n}{r} \cdot{ }^{n-1} C_{r-1}\right] \\
& =3\left[\sum_{r=0}^n(-1)^{r n} C_r\right]+5 n\left[\sum_{r=1}^n(-1)^r \cdot{ }^{n-1} C_{r-1}\right] \\
& =3(0)+5 n(0)=0+0=0
\end{aligned}
\end{aligned}
$$
$$
\begin{aligned}
& T_r=(-1)^r(3+5 r)^n C_r, r=0,1,2, \ldots, n \\
& \begin{aligned}
\therefore \quad S_n & =\sum_{r=0}^n(-1)^r(3+5 r)^n C_r \\
& =3 \sum_{r=0}^n(-1)^r \cdot{ }^n C_r+5 \sum_{r=0}^n(-1)^r r^n C_r \\
& =3\left[\sum_{r=0}^n(-1)^r \cdot{ }^n C_r\right]+5\left[\sum_{r=1}^n(-1)^r \cdot r \frac{n}{r} \cdot{ }^{n-1} C_{r-1}\right] \\
& =3\left[\sum_{r=0}^n(-1)^{r n} C_r\right]+5 n\left[\sum_{r=1}^n(-1)^r \cdot{ }^{n-1} C_{r-1}\right] \\
& =3(0)+5 n(0)=0+0=0
\end{aligned}
\end{aligned}
$$
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