Question

Find the coefficient of $x^{50}$ in ${(1+x)+2{(1+x)}^2+3{(1+x)}^3+....+1000(1+x)}^{1000}$.

Answer 1

Given $P\left(x\right)=(1+x)+2{(1+x)}^2+3{(1+x)}^3+....+999{(1+x)}^{999}+1000{(1+x)}^{1000}$

Let $S=(1+x)+2{(1+x)}^2+3{(1+x)}^3+....+999{(1+x)}^{999}+1000{(1+x)}^{1000}$ ....(1)

$(1+x)S=(1+x{)}^2+2{(1+x)}^3+3{(1+x)}^4+....+999{(1+x)}^{1000}+1000{(1+x)}^{1001}$ ....(2)
Subtracting (1) from (2), we get
$\Rightarrow xS=1000{(1+x)}^{1001}-\left[\right(1+x)+{(1+x)}^2+{(1+x)}^3+....+{(1+x)}^{999}+{(1+x)}^{1000}]$
Terms in the square brackets forms a G.P.

Using $S_n=\frac{a(r^n-1)}{r-1}$

$\Rightarrow xS=1000{(1+x)}^{1001}-\left[\right(1+x\left)\frac{{(1+x)}^{1000}-1}{1+x-1}\right]$
$\Rightarrow xS=1000{(1+x)}^{1001}-\left[\frac{{(1+x)}^{1001}-(1+x)}{x}\right]$

$\Rightarrow S=\frac{1000{(1+x)}^{1001}}{x}-\frac{{(1+x)}^{1001}}{x^2}+\frac{(1+x)}{x^2}$
Its clear from P(x), that the first two expansions can only have $x^{50}$.
So, coefficient of $x^{50}$ in $P\left(x\right)=\left(1000\right){{}^{1001}C}_{51}-{{}^{1001}C}_{52}$