Question

If $n$ is odd then value of $S=\frac{C_1}{C_0}-2\frac{C_2}{C_1}+3\frac{C_3}{C_2}-...$ upto $n$ terms is

• A. $\frac{1}{2}(n+1)$
• B. $\frac{1}{3}(n+1)$
• C. $\frac{1}{2}n(n+1)$
• D. none of these

Answer 1

The correct option is A $\frac{1}{2}(n+1)$

Writing the general term, we get
$(-1{)}^{r-1}r\frac{{}^n{C}_r}{{}^n{C}_{r-1}}$

$=(-1{)}^{-1}r\frac{n!(n-(r-1\left)\right)!(r-1)!}{n!(n-r)!.\left(r\right)!}$

$=(-1{)}^{r-1}r\frac{(n+1-r)}{r}$

$=(-1{)}^{r-1}(n+1-r)$

It is given $n$ is odd,
Hence summing, we get
$=n-(n-1)+(n-2)-\cdots (-1{)}^{n-1}(1)$

$=n-(n-1)+(n-2)-\cdots +\left(1\right)$

$=\frac{n+1}{2}$