MathLabs

Problem 2

Let n≥3n\ge3 be an integer, and let a2,a3,…,ana_2,a_3,\dots,a_n be positive real numbers such that a2a3⋯an=1a_2a_3\cdots a_n=1. Prove that (1+a2)2(1+a3)3⋯(1+an)n>nn(1+a_2)^2(1+a_3)^3\cdots(1+a_n)^n>n^n.
Step 3 of 4: Multiply the inequalities
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In plain words

The constants telescope from 2 through n.

∏k=2n(1+ak)k≥∏k=2nkk(k−1)k−1ak\prod_{k=2}^n(1+a_k)^k\ge\prod_{k=2}^n\dfrac{k^k}{(k-1)^{k-1}}a_k
Detailed analysis

Apply step 2 with x=akx=a_k and exponent kk, then multiply for k=2,…,nk=2,\dots,n. The product of the variable factors is 11 by step 1.