Question

The sum of the series ${\sum }_{r=1}^n(-1{)}^{r-1}{.}^n{C}_r(a-r)$ is equal to:

• A. n $2^{n-1}+a$
• B. 0
• C. a
• D. None of these

Answer 1

Answer: (C)

a

Let $S=S_1-S_2$

Where $S_1=\sum {(-1)}^{r-1}{n}_{C_r}\left(a\right)=a[\sum {(-1)}^{r-1}{n}_{C_r}]$

and $S_2=\sum {(-1)}^{r-1}r{n}_{C_r}$

$S_1=$ expansion of $\left\{a\{-[{(1+x)}^n-1]\}\right\}$ at $x=-1$

So, $S_1=1\times a=a$

$S_2=$ expansion of $\left\{\frac{d}{dx}\left({(1+x)}^n\right)\right\}$ at $x=(-1)$

So, $S_2=0$

So, $S=S_1-S_2=a-0=a$