Proof about commutator bracket

[ice109]

i've never really done a proof by induction but i would like to prove a statement about commutator relations so can you please check my proof:

claim: [A,B^n]=nB^{n-1}[A,B] if [A,B]=k\cdot I where A,B are operators, I is the identity and k is any scalar.

proof: [A,B^2] = [A,B]B+B[A,B] = 2B^{2-1}[A,B] where the B is brought out of the commutator by almost obvious identity and the last equality follows from the commutativity of the identity with any operator.

so for n=2 the equality is proven. we now assume that the equality holds for n-2 case and proceed to prove it holds for n case:

[A,B^n]=[A,B^2B^{n-2}]=[A,B^{2}]B^{n-2}+B^2[A,B^{n-2}]
=2B[A,B]B^{n-2}+B^2(n-2)B^{n-3}[A,B]
=2B^{n-1}[A,B]+(n-2)B^{n-1}[A,B]
=nB^{n-1}[A,B]

so how 'bout it? was successful?

 

[waht]

I think proof by induction goes something like this:

Show that the equality is true for n = 1

Assume the equality is true for n

Then show it is true for n+1

 

[ice109]

it's the same thing?