Help with induction proof , please. could someone please help me with the following induction proof.
I'm able to complete a somewhat similar proof where the RHS is equal to an algebraic expression ,but, can't get started with the RHS equal to a summation.
$$\sum_{i=1}^n i^3=\left(\sum_{i=1}^n i\right)^2$$
thanks
           ralph

Can I presume you know what "proof by induction" is?  You "get started" by showing that the statement is true for the 'first case' which, since the sum starts at n= 1, is evaluating both sides for n= 1.  The left side is just the single term $1^3= 1$.  The right side is single number, 1, squared, which is, of course, 1.  They are the same so you have proved the statement for n= 1.
Now, to complete the induction, we need to prove "if the statement is true for n= k then it is also true for n= k+ 1".  On the left side, going from n= k to n= k+ 1 means adding $(k+1)^3= k^3+ 3k^2+ 3k+ 1$.  Assuming that the statement is true for n= k means assuming that the sum of cubes is the same as $(\sum_{i=1}^k i)^2$ so adding $k^3+ 3k^2+ 3k+ 1$ gives $(\sum_{i=1}^k i)^2+ k^3+ 3k^3+ 3k+ 1$.  Is that the same as $(\sum_{i=1}^{k+1} i)^2$?
