To understand the logic of induction, I will just quote Wikipedia:
"It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one."
So let's do this with your problem:
Our conjecture (called ) is
1)First, let's check if the first term (when n = 1) is true:
2)Now, by letting n = k, we consider our conjecture for :
3)Then, by letting n = (k + 1), we consider the next case ( )
Notice how the blue part is actually . Thus, we may replace that series by the summation formula (all blue expressions are equal to each other):
4)So, being true necessarily implies that must also be true. Knowing that is true, we may use this (by letting k = 1), to conclude that is also true, and then (by replacing k = 2) conclude that is also true, repating this processes ad infinitum.
is true and the conjecture is proven.