What is Mathematical Induction?

Mathematical induction is a type of direct proof, where you can prove sequences or series. A good example of this is that we can prove 1 + 3 + 5 + .... + (2n-1) = n^2. There are 4 steps: 1. Prove the first case, or the n=1 case for this example. 2. Assume that the k-th case is true for any positive integer number k. 3. Using the assumption, prove that the (k+1)-th case. For this example we take n = k+1. 4. So we've just proved that if the k-th case is true then the (k+1)-th case must be true! So if the 1st case is true, then the 2nd case must be true. Then since the 2nd case is true, so must the 3rd case. This logic carries on and therefore we have proved what we wanted to prove for all integers!