induction exercise - having struggles with $1^3 + 2^3 + ... + (n+1)^3 = [(n(n+1))/2]^2$

Question

I'm trying to solve by induction that $1^3 + 2^3 + ... + (n+1)^3 = [(n(n+1))/2]^2$

However, I have a lot of trouble, and it must be said that I don't do a great deal of mathematics.

I keep getting $(k+1)^3 = (k+1)^2$ every time I try to solve it. Please point out where I am going wrong.

$[ (k(k+1))/2]^2 + (k+1)^3 = [ (k(k+1))/2 + (k+1) ]^2$

$(k^2(k+1)^2)/4 + (k+1)^3 = (k^2(k+1)^2)/4 + (k+1)^2$

Please help.

Answer 1

Please point out where I am going wrong.

$[ (k(k+1))/2]^2 + (k+1)^3 = [ (k(k+1))/2 + (k+1) ]^2$

$(k^2(k+1)^2)/4 + (k+1)^3 = (k^2(k+1)^2)/4 + (k+1)^2$

You did this on the RHS: $$\left(\frac{k(k+1)}2+(k+1)\right)^2 = \frac{k^2(k+1)^2}4 + (k+1)^2.$$ This is not correct. You have $(a+b)^2=a^2+2ab+b^2$, not $(a+b)^2=a^2+b^2$. So you should get $$\left(\frac{k(k+1)}2+(k+1)\right)^2 = \frac{k^2(k+1)^2}4 + k(k+1)^2 + (k+1)^2.$$ You may notice that $k(k+1)^2 + (k+1)^2= k(k+1)^2 + 1(k+1)^2 = (k+1)(k+1)^2 = (k+1)^3$. So you get exactly the equality you wanted.