Question

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