數學挑戰題 (8) --- Mathematical Induction (10分)

Question

Prove that the following inequality is true
by using Mathematical Induction :
(a^n + b ^n) / 2 > [(a + b) / 2]^n
Assume n is a natural number (n ≧ 2)
and a > b > 0

Answer 1

When n = 2,

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Maths/MIineq1.jpg

Therefore the inequality is true for n = 2.
Now, assuming that the inequality is true for n = k, where k is a positive integer ≧ 2, i.e.

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Maths/MIineq2.jpg

Here comes that the inequality is also true for n = k + 1.
By the first principle of M.I., the inequality is true for all positive integers n ≧ 2.