Question

Find the sum $3^n{C}_o-8^n{C}_1+{13}^n{C}_2-18{\times }^n{C}_3+...$

• A. $0$
• B. $1$
• C. $-1$
• D. $3$

Answer 1

The correct option is A $0$

The general term of the series is $T_r=(-1{)}^r(3+5r{)}^n{C}_r$
where r = 0, 1, 2, ..., n. Therefore, sum of the series is given by
$S=\sum _{r=0}^n(-1{)}^r(3+5r{)}^n{C}_r$
$=3(\sum _{r=0}^n(-1{)}^r{}^n{C}_r)$ + $5\left(\sum _{r=1}^n(-1{)}^r{n}^{n-1}{C}_{r-1}\right)$
$=3\left(\sum _{r=0}^n(-1{)}^r{}^n{C}_r\right)$ - $5n\left(\sum _{r=1}^n(-1{)}^{r-1}{}^{n-1}{C}_{r-1}\right)$
$=3(1-1{)}^n-5n(1-1{)}^{n-1}$
$=0$