Question

The value of the sum of the series $3^n{C}_0-8^n{C}_1+$ $13^n{C}_2-18^n{C}_3+\dots$ 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. General term of the series is given by,
$T_r=(-1{)}^r(3+5r{)}^n{C}_r,$ where $r=0,1,2,\dots ,n$
$\therefore S=\sum _{r=0}^n(-1{)}^r(3+5r){}^n{C}_r$
$\Rightarrow S=3\sum _{r=0}^n(-1{)}^r{}^n{C}_r+5\sum _{r=0}^n(-1{)}^{r}r{}^n{C}_r$
$\Rightarrow S=3\left(C_0-C_1+C_2-C_3+C_4\dots \right)$
$+5\left(-C_1+2C_2-3C_3+4C_4\dots \right)$
$\therefore S=0+0=0$