Mathematical Induction – Meaning
In general, the word Induction means the generalisation from particular cases or facts. In Mathematics, there are certain results or statements that are formulated in terms of n, where n is a positive integer. We may prove such general cases from Initial Base step to next one through the Mathematical Induction.
Mathematical Reasoning is the process of finding the proof for a certain mathematical statement by using logic and deductions. Inductive and deductive reasoning are two fundamental forms of reasoning for mathematicians. The formal theorems and proofs that we rely on today all began with these two types of reasoning.
- Mathematical Deduction : Deduction is drawing a conclusion from something known or assumed, normally used in almost every step in a mathematical argument.
- Mathematical Induction : Mathematical induction is a particular type of mathematical argument. It is most often used to prove general statements about Positive Integers.
To prove the formula, we may verify the statement for as many positive integral values of n as we like, but this process will not prove the formula for all values of n. We need is some kind of chain effect which will have the effect that once the formula is proved for a particular positive integer, the formula will automatically follow for the next positive integer, and the next indefinitely. Such chain effect is produced by Mathematical Induction.
Mathematical Induction Principles
Suppose there is a given statement P(n) involving the natural number n such that :
(i) The statement is true for n = 1, i.e., P (1) is true, and
(ii) If the statement is true for n = k (where k is some positive integer), then the statement is also true for n = k + 1, i.e., truth of P(k) implies the truth of P (k + 1). Then, P(n) is true for all natural numbers n.
Here, Property (i) is simply a statement of fact. When a statement is true for all n ≥4, we start from n = 4 and verify the result for n = 4, i.e., P(4).
Property (ii) is a conditional property. It does not assert that the given statement is true for n = k, but that if it is true for n = k, then it is also true for n = k +1. So, to prove that the property holds, prove the conditional proposition:
If the statement is true for n = k, then it is also true for n = k + 1.
This is referred to as the inductive step. The assumption that the given statement is true for n = k in this inductive step is called the inductive hypothesis.
Mathematical Induction – Example 1
Mathematical Induction – Example 2
Mathematical Induction – Example 3
Mathematical Induction – Example 4