First you prove that it holds for n=1.
Now you assume that it holds for k, and prove that it holds for k+1.
If you write the right side with common denominator:
And you´re there.
(*) For all natural numbers n, 1 + 2 + 3 + 4 + ... + n = (n)(n+1)/2
Assume for k:
... + k = (k)(k+1)/2
Assume for k+1:
... + k+1 = (k+1)(k+2)/2
I believe I need to increment the series to actually create a valid test:
... k + k+1 = (k+1)(k+2)/2
Hmm.. the sides don't seem to match, where did I go wrong?