Coefficient of $x^3$ in $(1+x^2)(1+x)^{100}$
Coefficient of $x^{10}$ in $(1+x)^{10}(1-x)^{10}$
Coefficient of $x^n$ in $\dfrac{2+x}{2-x}$.
Any help on these would be appreciated.
Coefficient of $x^3$ in $(1+x^2)(1+x)^{100}$
Coefficient of $x^{10}$ in $(1+x)^{10}(1-x)^{10}$
Coefficient of $x^n$ in $\dfrac{2+x}{2-x}$.
Any help on these would be appreciated.
There are two ways to obtain $x^3$ in $(1+x^2)(1+x)^{100}$: when you multiply $1$ on the first binomial by the $x^3$-term in the expansion of $(1+x)^{100}$, and when you multiply $x^2$ by the $x$-term in the expansion of $(1+x)^{100}$.
You obtain $x^{10}$ by multiplying the $x^i$ term in the expansion of $(1+x)^{10}$ with the $x^{10-i}$ term in the expansion of $(1-x)^{10}$, for $i=0,1,\ldots,10$.
Expand $(2-x)^{-1}$ and figure out the $x^n$ and $x^{n-1}$ terms.
Follow suggestion given by Arturo. The one shown above is just to clear things for you.
Do similarly approach the other answers