Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 6.1 National olympiad Prove it

16. Given the sequence $a_{1}, a_{2}, \cdots, a_{n}$ satisfies $a_{1}=\sqrt{2}, a_{2}=2, a_{n+1}=a_{n} a_{n-1}^{2}, n \geqslant 2$ , prove: $\left(1+a_{1}\right)\left(1+a_{2}\right) \cdots\left(1+a_{n}\right)<(2+\sqrt{2}) a_{1} a_{2} \cdots a_{n}$ . (2003 Baltic Way Mathematical Competition)

Solution

16. By induction, we can prove that $a_{n}=2^{2 n-2}$ , so we only need to prove $\left(1+a_{2}\right)\left(1+a_{3}\right) \cdots(1+$ $\left.a_{n}\right)<2 a_{2} a_{3} \cdots a_{n}$ .

The left side of the inequality $=2^{2 n-1}$ , the right side $=2^{2 n-5}-1$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.