How to find coefficient of in a binomial expression

Question

I have to find coefficient of $x^3$ in $(x + 1)^n + (x + 1)^{n - 1}(x + 2) + (x + 1)^{n - 2}(x + 2)^2 + \ldots + (x + 2)^n$ and I cannot get a starting point as to how to solve this.

Answer 1

We can start by simplifying the sum as follows but also note that we assume here $n\ge 3$, $$ \displaystyle \sum_{k=0}^n (x+1)^{n-k}(x+2)^{k} = (x+2)^{n+1}-(x+1)^{n+1}$$ This can be derived with the help of finite geometric series. Now the coefficient of $x^r$ in $(x+a)^p$ is, $$\displaystyle [x^r](a+x)^p = \binom{p}{r}a^{p-r}$$ Using this the coefficient of $x^3$ is, $$ \mathfrak{L} = \binom{n+1}{3}\left(2^{n-2}-1\right)$$ This is the required answer.