Exam-Style Problem
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FM November 2021 p11 q03
4276
The sequence of real numbers \(a_1, a_2, a_3, \ldots\) is such that \(a_1 = 1\) and
\(a_{n+1} = \left( a_n + \frac{1}{a_n} \right)^3.\)
(a) Prove by mathematical induction that \(\ln a_n \geq 3^{n-1} \ln 2\) for all integers \(n \geq 2\).
[You may use the fact that \(\ln \left( x + \frac{1}{x} \right) > \ln x\) for \(x > 0\).]
(b) Show that \(\ln a_{n+1} - \ln a_n > 3^{n-1} \ln 4\) for \(n \geq 2\).
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