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