06/04/2020 (2011: National Sprint Round, Problem 14)

Q: In the sequence 8, 25, 76, 229..., each term after the first term is b more than a times the preceding term. What is the value of a + b?

A: 4

We set up an equation with f(x) being a value in the series and f(x-1) being the previous number: f(x) = af(x-1) + b -->

25 = a(8) + b AND 76 = a(25) + b

We subtract the first equation from the second and get: 51 = 17a

We divide both sides by 17 and get: 3 = a

We then plug in the values above into the new equation, f(x) = 3f(x-1) + b, and get: 25 = 3(8) + b

That simplifies to 25 = 24 + b, which makes 1 = b

That means a + b = 3 + 1 = 4