Commutator proof

Question

I have a proof in my book I dont fully understand. The author is proving that if $[A,B]=1$ then $[A,B^n]=nB^{n-1}$. The proof is really short, it is only one line of equations, but I dont understand it. Here it is:$$[A,B^{n+1}]=AB^{n+1}-B(AB^n-nB^{n-1})$$ My question is, how did this $AB^n-B^nA$ become that ^^^?

Answer 1

This is a proof by induction. The base case is $[A, B] = I = 1 B^{1-1}$. For the induction step, assuming it is true for $n=k$, i.e. $[A,B^k] = k B^{k-1}$, we compute: $$ [A, B^{k+1}] = A B^{k+1} - B^{k+1} A = A B^{k+1} - B^k A B + B^k A B - B^{k+1} A$$ Now $$A B^{k+1} - B^{k} A B = (A B^k - B^k A) B = k B^{k-1} B = k B^k$$ and $$B^kAB - B^{k+1} A = B^k (AB - BA) = B^k I = B^k$$ so the sum is $(k+1) B^k$, and the statement is true for $n=k+1$.