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Q.
The value of the sum of the series $3^{n} C_{0}-8^{n} C_{1}+$ $13^{n} C_{2}-18^{n} C_{3}+\ldots$ upto $(n+1)$ terms is
Binomial Theorem
Solution:
Let $S$ denotes the sum of the series. General term of the series is given by,
$T_{r}=(-1)^{r}(3+5 r)^{n} C_{r}, $ where $r=0,1,2, \ldots, n$
$\therefore S=\displaystyle\sum_{r=0}^{n}(-1)^{r}(3+5 r){ }^{n} C_{r}$
$\Rightarrow S=3 \displaystyle\sum_{r=0}^{n}(-1)^{r}{ }^{n} C_{r}+5 \displaystyle\sum_{r=0}^{n}(-1)^{r} r{ }^{n} C_{r} $
$\Rightarrow S=3\left(C_{0}-C_{1}+C_{2}-C_{3}+C_{4} \ldots\right)$
$+5\left(-C_{1}+2 C_{2}-3 C_{3}+4 C_{4} \ldots\right) $
$\therefore S=0+0=0$