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a_n is a sequence with a_0=1, a_1=3, a_{n+2}=...

Source: Vietnamese TST 2011 P4

April 27, 2011
floor functioninductionnumber theory proposednumber theory

Problem Statement

Let ⟨an⟩n≥0\langle a_n\rangle_{n\ge 0} be a sequence of integers satisfying a0=1,a1=3a_0=1, a_1=3 and an+2=1+⌊an+12an⌋  ∀n≥0.a_{n+2}=1+\left\lfloor \frac{a_{n+1}^2}{a_n}\right\rfloor \ \ \forall n\ge0. Prove that an⋅an+2−an+12=2na_n\cdot a_{n+2}-a_{n+1}^2=2^n for every natural number n.n.
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