Question

Use mathematical induction to prove

Answer 1

`Prove\ that\ the\ assumption \is \true for\ n=1\\1^3=(1^2(1+1)^2)/(4)\\ 1=(4)/(4)=1\\`

Formula works when n=1

Assume the formula also works, when n=k.

Prove that the formula works, when n=k+1

`1^3+2^3+3^3...+k^3+(k+1)^3=((k+1)^2(k+2)^2)/(4) \\(k^2(k+1)^2)/(4)+(k+1)^3=((k+1)^2(k+2)^2)/(4) \\(k^2(k^2+2k+1))/(4)+(k+1)^3=((k^2+2k+1)(k^2+4k+4))/(4) \\(k^4+2k^3+k^2)/(4)+k^3+3k^2+3k+1=(k^4+4k^3+4k^2+2k^3+8k^2+8k+k^2+4k+4)/(4)\\\\(k^4+2k^3+k^2)/(4)+k^3+3k^2+3k+1=(k^4+6k^3+13k^2+12k+4)/(4)\\(k^4+2k^3+k^2)/(4)+(4k^3+12k^2+12k+4)/(4)=(k^4+6k^3+13k^2+12k+4)/(4)\\(k^4+6k^3+13k^2+12k+4)/(4)=(k^4+6k^3+13k^2+12k+4)/(4)\\`

Since the formula has been proven with n=1 and n=k+1, it is true.
`\square`