Question

Let x > -1 then statement (1+x)n > 1+nx is true for

Answer 1

Correct option: (c) ∀ n >1 and x ≠ 0

Explanation:

For n = 2,

P(2) : (1 + x)2 = 1 + 2x + x2 > 1 + 2x

as x ≠ 0.

Assume that,

P(k) : (1 + x)k > 1 + kx (1)

for some k ∈ N, k > 1.

As x > – 1, multiplying both

sides by (1 + x)

∴ from (1),

(1 + x)k+1 > (1 + kx) (1 + x)

∴ (1 + x)k+1 > 1 + (k + 1)x + kx2

∴ (1 + x)k+1 > 1 + (k + 1)x ------ (kx2 > 0)

∴ P(k + 1) is true

∴ by principle of mathematical induction, P(n) is true for all n > 1 provided x ≠ 0.