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Interesting sequence for Juniors

Source: Turkey JBMO TST 2024 P6

May 13, 2024
algebraSequence

Problem Statement

Let (an)n=0∞{(a_n)}_{n=0}^{\infty} and (bn)n=0∞{(b_n)}_{n=0}^{\infty} be real squences such that a0=40a_0=40, b0=41b_0=41 and for all n≥0n\geq 0 the given equalities hold. an+1=an+1bnandbn+1=bn+1ana_{n+1}=a_n+\frac{1}{b_n} \hspace{0.5 cm} \text{and} \hspace{0.5 cm} b_{n+1}=b_n+\frac{1}{a_n} Find the least possible positive integer value of kk such that the value of aka_k is strictly bigger than 8080.
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