Question

The value of the sum of the series $3^n{C}_0-8^n{C}_1+{13}^n{C}_2-{18}^n{C}_3+...$ upto $(n+1)$ term is

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

Answer 1

Answer: (A)

$0$

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 $r=0,1,2,...,n$

$\therefore S=\sum _{r=0}^n{(-1)}^r{(3+5r)}^n{C}_r$

$\Rightarrow S=3\sum _{r=0}^n{(-1)}^r{.}^n{C}_r+5\sum _{r=0}^n{(-1)}^r{.}^n{C}_r$

$\Rightarrow S=3(C_0-C_1+C_2-C_3+C_4...)+5(-C_1+2C_2-3C_3+4C_4...)\therefore S=0+0=0$