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(1 + \sqrt5)^n =\sqrt{a_n} + \sqrt{a_n+4^n}

Source: 2010 Saudi Arabia IMO TST V p2

December 28, 2021

number theoryalgebraSequence

Problem Statement

a) Prove that for each positive integer

n

there is a unique positive integer

a_n

such that

(1 + \sqrt5)^n =\sqrt{a_n} + \sqrt{a_n+4^n} .

b) Prove that

a_{2010}

is divisible by

5\cdot 4^{2009}

and find the quotient

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