We prove $\sum_{i=1}^{n} x^3 = (1+2+\cdots+n)^2$. We observe that this expression is true for $ n=1$. Now assume this is already true; we prove it for $\sum_{i=1}^{n+1} x^3 = (1+2+\cdots+n+n+1)^2$. We note that the expression in the inside of the parentheses is equal to $\frac {(n+1)(n)}{2}$. But when we add an $n+1$ term, the inside is equal to$\frac{(n+1)(n)}{2} + n + 1$, or $\frac {(n+1)(n+2)}{2}$. QED
First of all, did I prove this correctly? Next, instead of just $x^3$ I can write $ x^n $ and it will still hold true. Isn't this a contradiction?
As pointed out by the comments, there definitely a typo. Apart from that you haven't really proven anything, you've just rewritten the right hand side a little.
I assume what you want to prove is: $$ \sum_{k=1}^{n} k^3 = \left(\sum_{k=1}^{n} k\right)^2 $$
To prove that by induction, you start by proving the base case ($n=1$) and that's easy. Then you look at $$ \sum_{k=1}^{n+1} k^3 $$ and prove (using the induction hypothesis, i.e. the result for the sum going to $n$) that it is equal to $$ \left(\sum_{k=1}^{n+1} k\right)^2 $$
In other words fill in something for the dots in (you might want to manipulate the expression in other ways before using the induction hypothesis: $$ \sum_{k=1}^{n+1} k^3 =\sum_{k=1}^{n} k^3+(n+1)^3 =\left(\sum_{k=1}^{n} k\right)^2+(n+1)^3 =\cdots =\left(\sum_{k=1}^{n+1} k\right)^2 $$
You might also want to start with the right hand side for $n+1$ and rewrite it to the left hand side$\ldots$