Question

Exercise 2.4. Prove the two statements below:Use nd ueTion 1. For every integer n 2 3, the inequality n2 2n +1 holds. Hint: You can prove this by induction if you wish, but alternatively, you can prove directly, without induction.) 2. For every integer n 2 5, the inequality 2" n holds. (Hint: Use induction and the inequality in the previous part of the exercise.)

Answer 1

We prove n^2 > 2n + 1: n greaterthanorequalto 3 for n = 3, 3^2 = 9 > 2 times 3 + 1 = 7 this stat inequlity is true for n = 3 Assume that it is true for n = -k k^2 > 2k + 1 we prove that it is true for n = k + 1 L.H.S = (k + 1)^2 = k^2 + 2k + 1 but clear k^2 > 2 for n greaterthanorequalto 3 implies k^2 + 2k > 2 + 2k implies k^2 + 2k + 1 > 3 + 2k implies k^2 + 2k = 1 > 2k + 2 + 1 implies (x + 1)^2 > 2(x + 1) + 1 so the above inequality is true for n = k + 1 Therefore by method of induction it is there for all n greaterthanorequalto 3 \r\n we prove 2^n > n^2 for n greaterthanorequalto 5 for n = 5, 2^5 = 32 > 5^n = 25 Therefore This inequality is true for n = 5 Assume that it is true for n = k i.e 2^k > k^2 implies 2.2^k > 2k^2 implies 2^k + 1 > 2k^2