Should say ASSUME n = k is true.NOTE: (1+...+k)=(k(k+1))/2 by sum of Arithmetic Progression.
First case for n=1 is trivial
Let n=k:
1^3+...+k^3=(1+...+k)^2
Let n=k+1:
LHS=1^3+...+k^3+(k+1)^3
=(1+...+k)^2+(k+1)^3
=k^2(k+1)^2/4+(k+1)^3
=RHS
RHS=((1+...+k)+k+1)^2
=(1+...+k)^2+2(k+1)(1+...+k)+(k+1)^2
=k^2(k+1)^2/4+(k+1)(k(k+1)+(k+1))
=k^2(k+1)^2/4+(k+1)^2.(k+1)
=k^2(k+1)^2/4+(k+1)^3
Above post is wrong as his case doesn't work for n=2 and has misinterpreted the question:
ie 1^3+2^3 =/= (1+2)^3
Yeah, sorry. Usually i say: Assume < statement> (for n=k) but i didn't think it was necessary as i'm just writing out the proof for a person and not formally doing it in a test.Should say ASSUME n = k is true.