Prove ∑r^3 = 1/4 n^2(n+1)^2

Proof by induction

Base Case when n=1 

LHS = 1^3=1 RHS= 1/4(1^2)(1+1)^2=1/4(1)(2^2)=1/4(4)

Assume true for n=k 

so  ∑r^3= 1/4k^2(k+1)^2

For n=k+1 

∑r^3 = ∑k terms + (k+1)^3 = 1/4(k^2)(k+1)^2 + (k+1)^3

= 1/4(k^2)(k+1)^2 + (k+1)^2(k+1)

=1/4(k^2+4k+4)(k+1)^2 

Completing the square k^2+ 4k + 4 = (k+2)^2

=1/4(k+2)^2(k+1)^2

Same form as above with n replaced by k+1

Therefore it is true for n=k+1 if true for n=k but true for n=1 so true for n=2 and so on.