Principle of Mathematical Induction

CBSE Class 11 Mathematics

Revision Notes
Chapter-4
PRINCIPLE OF MATHEMATICAL INDUCTION

One key basis for mathematical thinking is deductive reasoning. In contrast to deduction, inductive reasoning depends on working with different cases and developing a conjecture by observing incidences till we have observed each and every case. Thus, in simple language we can say the word ‘induction’ means the generalisation from particular cases or facts.
Statement: A sentence is called a statement, if it is either true ot false.
Motivation: Motivation is tending to initiate an action. Here Basis step motivate us for mathematical induciton.
Principle of Mathematical Induction: The principle of mathematical induction is one such tool which can be used to prove a wide variety of mathematical statements. Each such statement is assumed as P(n) associated with positive integer n, for which the correctness for the case n = 1 is examined. Then assuming the truth of P(k) for some positive integer k, the truth of P (k+1) is established.
Working Rule:
Step 1: Show that the given statement is true for n = 1.
Step 2: Assume that the statement is true for n = k.
Step 3: Using the assumption made in step 2, show that the statement is true for n = k + 1. We have proved the statement is true for n = k. According to step 3, it is also true for k + 1 (i.e., 1 + 1 = 2). By repeating the above logic, it is true for every natural number.