AIMO 2015 Q9

Number Theory Level 4

A sequence is formed by the following rules:

$s_1 = a, s_2 = b$ and $s_{n+2} = s_{n+1} + (-1)^n s_n$ for all $n \geq 1$ .

If $a=3$ and $b < 1000$ , what is the largest integral value of $b$ for which 2015 is a member of the sequence?

The answer is 253.

1 solution

Anand Raj
Jun 12, 2019

If the second term is $b$ , then after writing a few terms of the series, I found that the terms were of the form: $b$ , $b-3$ , $2b-3$ , $3b-3$ , $5b-6$ , $8b-9$ and so on. Equating these values to 2015 gives the first integer answer for $8b-9=2015\Rightarrow b=253$ .

FunFact: This sequence follows a Fibonacci pattern except for its second term, $b-3$ .

AIMO 2015 Q9

The answer is 253.

1 solution

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