Class 11 Maths Mathematical Induction Example

Example: Prove that 2n  > n for all positive integers n

Solution:  Let P(n): 2n > n

Step 1: When n =1, 21 >1. Hence P(1) is true.

Step 2: Assume that P(k) is true for any positive integer k, i.e., 2k > k ... (1)

Step 3: We shall now prove that P(k +1) is true whenever P(k) is true.

Multiplying both sides of (1) by 2, we get

2* 2k > 2*k

i.e., 2 k + 1 > 2k

or,  2 k + 1 >  k + k

or, 2 k + 1 >  k + 1    (since  k>1)

Therefore, P(k + 1) is true when P(k) is true. Hence, by principle of mathematical induction, P(n) is true for every positive integer n

 

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