1. ## Mathematical induction

(*) 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?

2. First you prove that it holds for n=1.

$1 = (1)(1+1)\frac{1}{2} = 1 \; \mbox{ ok! }$

Now you assume that it holds for k, and prove that it holds for k+1.

$1+2+3+ \dots k + (k+1)=k(k+1)\frac{1}{2}+(k+1)$

If you write the right side with common denominator:

$\mbox{Right side: } \frac{1}{2}(k(k+1)+(k+1)) = \frac{1}{2}(k+1)(k+2)$

And you´re there.