MathLabs

Problem 6

Let a1,…,ana_1,\ldots,a_n be positive numbers and let 0<q<10<q<1. Find real numbers b1,…,bnb_1,\ldots,b_n such that (1) ak<bka_k<b_k for every kk; (2) q<bk+1/bk<1/qq<b_{k+1}/b_k<1/q for k=1,…,n−1k=1,\ldots,n-1; and (3) ∑k=1nbk<1+q1−q∑k=1nak\sum_{k=1}^n b_k<\frac{1+q}{1-q}\sum_{k=1}^n a_k.
Step 2 of 4: Verify the first condition
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In plain words

Every index receives its own value plus positive contributions from elsewhere.

br>arb_r>a_r
Detailed analysis

The i=ri=r term equals ara_r, while every i≠ri\ne r term is positive. Hence br>arb_r>a_r.