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For which values of n does the inequality 4ⁿ/(n+1) < (2n)!/(n!)² hold, if P(n) denotes this statement?

1. n ≥ 1
2. n < 0
3. n > 0
4. n ≥ 2

Correct answer: n ≥ 2

Solution

The right side is C(2n,n). At n=1, 4/2 = 2 is not less than C(2,1) = 2, so it fails. For n=2, 16/3 < 6, and it continues to hold for every larger n, so the statement is true for n >= 2.

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