Question

Prove this using PMI for all n element of N:1^3+2^3^3+.............+n^3=(n(n+1)÷2)^2

Answer 1

:

Base case: n=1

L.H.S:13=1

R.H.S:(1)2=1

Therefore it's true for n=1.

I.H.: Assume that, for some k∈N, 13+23+...+k3=(1+2+...+k)2.

Want to show that 13+23+...+(k+1)3=(1+2+...+(k+1))2

13+23+...+(k+1)3

=13+23+...+k3+(k+1)3

=(1+2+...+k)2+(k+1)3 by I.H.

Annnnd I'm stuck. Not sure how to proceed from here on.

Answer 2

Answer:

Prove this using PMI for all n element of N:1^3+2^3^3+.............+n^3=(n(n+1)÷2)^2