Question

The coefficient of $x^r(0\le r\le n-1)$ in the expansion of
$E=2^n+2^{n-1}(x+2)+2^{n-2}{(x+2)}^2+....+{(x+2)}^n$

• A. ${{}^{n+1}C}_{r+1}{2}^{n-r}$
• B. ${{}^{n}C}_r{2}^{n-r}$
• C. ${{}^{n}C}_r{2}^r$
• D. none of these

Answer 1

Answer: (A)

${{}^{n+1}C}_{r+1}{2}^{n-r}$

$2^n$+$2^{n-1}$(x+2)+ $2^{n-2}$.${(x+2)}^2$+...........................+${(x+2)}^n$

$2^n$(1+$\left(\frac{x+2}{2}\right)$+ $($$\frac{x+2}{2}{)}^2$+...........................+$(\frac{x+2}{2}{)}^n$$)$

now its sum come out as

$2^n$( $\frac{({1+\frac{x}{2})}^{n+1}-1}{1+\frac{x}{2}-1}$$)$

$2^{n+1}$( $\frac{({1+\frac{x}{2})}^{n+1}-1}{x}$$)$

So the coefficent of it will be found by finding coefficent of $x^{r+1}$ in the expansion of ${(1+\frac{x}{2})}^{n+1}$ as ther is x in denominator

hence coefficent is

$2^{n+1}$.${}^{n+1}{C}_{r+1}$${\left(\frac{1}{2}\right)}^{r+1}$

$2^{n-r}$.${}^{n+1}{C}_{r+1}$