Question

The value of the sum of the series $3 \cdot{ }^{n} C_{0}-8 \cdot{ }^{n} C_{1}+13^{n} C_{2}-18 \cdot{ }^{n} C_{3}+\ldots$ upto $(n+1)$ terms is

• A. 0
• B. $3^{n}$
• C. $5^{n}$
• D. None of these

Answer 1

Answer: (A)

Let $S$ denotes the sum of the series. The general term of the given series is
$T_{r}=(-1)^{r}(3+5 r)^{n} C_{r}(n t h$ term of $A P)$
$\therefore S=\displaystyle\sum_{r=0}^{n}(-1)^{r}(3+5 r)^{n} C_{r}$
$=3 \displaystyle\sum_{r=0}^{n}(-1)^{r n} C_{r}+5 \displaystyle\sum_{r=0}^{n}(-1)^{r} r^{n} C_{r}$
$= 3(C_{0}-C_{1}+C_{2}-C_{3}+C_{4}-\ldots ..$
$+(-1)^{n} \cdot C_{n})+5(-C_{1}+2 C_{2}-3 C_{3}$
$\left.+4 C_{4}-\ldots+(-1)^{n} \cdot n \cdot C_{n}\right)$
$\Rightarrow S=0+0=0$