Question

The sum of the series $3^n{C}_0-8^n{C}_1+{13}^n{C}_2-{18.}^n{C}_3+---$ is

• A. 1
• B. -5
• C. -27

Answer 1

$S={\sum }_{r=0}^n(-1{)}^r(3+5r).n_{C_r}$
$=3\left({\sum }_{r=0}^n\right(-1{)}^r.n_{C_r})+5\left({\sum }_{r=0}^n\right(-1{)}^r.n.n-1_{C_{r-1}})$
$=3(1-1{)}^n+5n(1-1{)}^{n-1}=0$