Question

The coefficient of $x^r(0\ge r\ge n-1)$ in the expansion of
$(x+2{)}^{n-1}+(x+2{)}^{n-2}(x+1)+(x+2{)}^{n-3}(x+1{)}^2+.....+(x+1{)}^{n-1}$ is ${}^n{C}_r(2^{n-r}-1)$

Answer 1

As the above the given series is a G. P. of n terms where
$a=(x+2{)}^{n-1}andx=\frac{x+1}{x+2}$
$S=\frac{a(r^n-1)}{r-1}=(x+2{)}^{n-1}[{\left(\frac{x+1}{x+2}\right)}^n-1]+[\frac{x+1}{x+2}-1]$
$\frac{(x+1{)}^n-(x+2{)}^n}{-1}=(2+x{)}^n-(1+x{)}^n$
$x^r$ will occur in $T_{r+1}$ are both .
${}^n{C}_r,2^{n-r}\cdot x^r-{}^n{C}_r{x}^r$
$\therefore coefficientof{x}^{r}is{}^n{C}_r(2^{n-r}-1)$