|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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