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2007 Guts #22: Recursive Sequences GCD

Source:

June 22, 2012

number theorygreatest common divisor

Problem Statement

The sequence

\{a_n\}_{n\geq 1}

is defined by

a_{n+2}=7a_{n+1}-a_n

for positive integers

n

with initial values

a_1=1

and

a_2=8

. Another sequence,

\{b_n\}

, is defined by the rule

b_{n+2}=3b_{n+1}-b_n

for positive integers

n

together with the values

b_1=1

and

b_2=2

. Find

\gcd(a_{5000},b_{501})

.

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