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.