Mathematical Induction Problems

Mathematical Induction is an algebraic method to show that a statement or a postulate is true by using the contradiction technique. According to this method we assume a hypothesis in the starting which has to be proved wrong by the end of complete calculation. So to prove your induction to be true, the considered statement must be satisfied by at least one number. Each step of Mathematical Induction problems are to be proved using mathematical formulas.

Let us consider an example of mathematical induction as H² >= 2H where H = 2, 3, 4 and so on. Mathematical induction steps for this problem: To start with consider your induction hypothesis that you want to prove. If H² >= 2H is true then it should be true for H = k also, where k = 2, 3, 4 and so on. So, we can write above equation as:

 k² >= 2k. Therefore if it is true for H = k we must now prove our hypothesis true for H = k + 1 also. This is the step where you actually prove your hypothesis wrong by contradiction by performing certain mathematical calculations. So, problem can be written as:

 i = k + 1 then (k + 1)² >= 2 (k + 1) for every (k + 1) = 2, 3, 4 and so on.
(k + 1)² >= 2 (k + 1),
k² + 2k + 2 >= 2k + 2,
We know k = i and i² = 2i so k² = 2k.
2k + 2k + 2 >= 2k + 2,
We know 2k > 1 because k > 1,
2k + 2k + 1 > 2k + 2,

Thus we see here that left side comes out to be greater than right side which proves our induction by contradicting situation.