YLL討論網

證明

2n+1小於等於2ⁿ

for all integer n大於等於3

Let P(n) be the statement that "2ⁿ≥2n+1"
For n=3, 2³ = 8 ≥ 7 = 2(3)+1
i.e. P(3) is true.
asume P(k) is true ∃k∈Z, k≥3, i.e. 2^k≥2k+1
For n=k+1, 2^(k+1) = 2(2^k) ≥ 2(2k+1) = 4k+2 = 2(k+1)+2k > 2(k+1)+1
i.e. P(k+1) is also true.
By M.I., P(n) is true ∀n∈Z, n≥3