AMO-Bench 2025 Problem 5

Answer

The sequence $\{a_n\}_{n=1}^{\infty}$ consists of positive terms, with $a_1=7$ , $a_2=2$ , and satisfies the recurrence relation $8a_{n+2}^4=3+4a_{n+1}+a_n (n\in\mathbb{N}^*).$ Find the general term formula for this sequence.

Dataset text AMO-Bench. Open licence.

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$\boxed{\dfrac{(2+\sqrt{3})^{2^{2-n}}+(2-\sqrt{3})^{2^{2-n}}}{2}}$

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