Show that $$E(X^k)=npE((Y + 1)^{k-1})$$ where $X\sim\mathrm{Bin}(n,p)$ and $Y \sim\mathrm{Bin}(n-1,p)$.
I am looking for suggestions on where to start? Or any resources someone may have. I am not even sure where to begin. I have the pdf, cdf, variance, mean and moment generating function of the binomial distribution given to me but I am not sure how to use those.
Hint: Fill up details in the following:
With $\;X\sim B(n,p)\;$ and using the identity $\;i\binom ni=n\binom{n-1}{i-1}\;$:
$$E(X^k)=\sum_{i=0}^n i^k\binom ni p^i(1-p)^{n-i}=\sum_{\color{red}{i=1}}^n i^k\binom ni p^i(1-p)^{n-i}=$$
$$=np\sum_{i=1}^n i^{k-1}\binom{n-1} {i-1} p^{i-1}(1-p)^{n-i}\stackrel{m:=i-1}=np\sum_{m=0}^n (m+1)^{k-1}\binom{n-1} m p^m(1-p)^{n-1-m}=$$
$$=np\,E\left[(Y+1)^{k-1}\right]\;\;,\;\;\;\text{with}\;\;Y\sim B(n-1,p)$$