An over-ambitious AP

Algebra Level 4

Let \({a_{i}},0 \le i \le n\) be an arithmetic progression (AP) such that \(a_{0} = 16\) and \(a_{j} > 0\) for all \( j \ge 0\).

Also,every three consecutive terms of this AP satisfy the inequality \[\displaystyle \dfrac{1}{a_{n}} \ge \dfrac{1}{\sqrt{a_{n+1} \cdot a_{n-1}}}.\]

Find the value of \(\displaystyle \sum_{j=0}^{99} \dfrac{\sqrt[4]{a_{j}}}{a_{j}} \).

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