Prove that 1 1 + a 1 </msub> +

Nasir Kim

Nasir Kim

Answered question

2022-05-22

Prove that 1 1 + a 1 + a 1 a 2 + 1 1 + a 2 + a 2 a 3 + + 1 1 + a n 1 + a n 1 a n + 1 1 + a n + a n a 1 > 1.
I find it hard to use any inequalities here since we have to prove > 1 and most inequalities such as AM-GM and Cauchy-Schwarz use 1. On the other hand it seems that if I can prove that each fraction is > 1 that might help, but I am unsure.

Answer & Explanation

Vitulloh0

Vitulloh0

Beginner2022-05-23Added 8 answers

Let a i = x i + 1 x i , where x n + 1 = x 1 , x n + 2 = x 2 and all x i > 0. Hence,
i = 1 n 1 1 + a i + a i a i + 1 = i = 1 n x i x i + x i + 1 + x i + 2 > i = 1 n x i x 1 + x 2 + . . . + x n = 1

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