MathDB

13

Part of 2005 National High School Mathematics League

Problems(1)

Sequence

Source: 2005 National High School Mathematics League, Exam One, Problem 13

3/20/2020
Define sequence (an)(a_n): a0=1,an+1=7an+45an2−362,n∈Na_0=1,a_{n+1}=\frac{7a_n+\sqrt{45a_n^2-36}}{2},n\in\mathbb{N}. (a) Prove that for all n∈Nn\in\mathbb{N}, ana_n is a positive integer. (b) Prove that for all n∈Nn\in\mathbb{N}, anan+1−1a_na_{n+1}-1 is a perfect square.