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in Mathematics by (71.2k points)

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

1 Answer

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Best answer

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.

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