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a_{n + 1} =\sqrt{a_n^2 + 1}

Source: 2010 Saudi Arabia Pre-TST 2.3

December 28, 2021
irrational numberSequencealgebra

Problem Statement

Let a0a_0 be a positive integer and an+1=an2+1a_{n + 1} =\sqrt{a_n^2 + 1}, for all n≥0n \ge 0.
1) Prove that for all a0a_0 the sequence contains infinitely many integers and infinitely many irrational numbers.
2) Is there an a0a_0 for which a2010a_{2010} is an integer?
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