Mathematical induction describe as a domino cards or climbing a ladder. In domino cards, if the first card falls, then the second falls too and so does the third, fourth and finally all of cards fall.
Climbing a ladder is like domino cards. If the first step is passed, then going to second step, third step and finally all of step ladder is passed.
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Principle of mathematical induction
Mathematical induction is one of proof technique in mathematics. If a statement is true based on mathematical induction, it means that it is also true for all of natural numbers.
Mathematical statement that can be proved by using mathematical induction is like a sequence statement P(n) for natural numbers (n=1,2,3,…) of infinitely P(1), P(2), P(3), ….
Principle of using mathematical induction consists of two steps. There are base and induction step.
1. Base step (it is also called basis step) prove statement that n = 0 (or based on the problem’s condition) is true. It is also often use n = 1.
If the problem said that n = a, so the base step is n = b that b>a (a and b is natural numbers). Proving the statement must be independent, it is not depends on other condition or other problems.
2. Induction step proves that for every n natural numbers, if given n=k is true then n=k+1 is also true for the problems.
After all of steps are true then the statement of the problem is true.
Based on the principle, there are simple steps that will make it easily.
- Prove that n=1 is true for the problem.
- Assume that n=k is true then proves n=k+1. Be careful doing this second step, it needs your trick and your logic.
Mathematical induction examples
Generally, there are three types of mathematical induction problems.
1. Sum of consecutive natural numbers
Prove that 1 + 3 + 5 + … + (2n-1) = n2 for n is positive integer.
Prove that 52n+3n-1 can be divided by 9.
Prove that 3n-1 is multiple of 2 for n = 1, 2, …
Prove that 4n < 2n for n>5.