Q. Solving a Quadratic Recurrence Relation: Finding Coefficients a, b, and c

[odolwa99]

Homework Statement

Dang it! More recurrence relation problems, but this time it’s due to a quadratic equation.

Q. In the sequence u₁, u₂, u₃,…,u_n,
u₁ = 0, u₂ = 3, u₃ = 12 and u_n = a + bn + cn²
Find the values of a, b and c.

Homework Equations

Provided at back of book…
Answer: a = 3, b = -6, c = 3

The Attempt at a Solution

Attempt:
If n = 1 then u₁ = a + b(1) + c(1)² = 0
= a + b + c = 0
If n = 2 then u₂ = a + b(2) + c(2)² = 3
= a + 2b + 4c = 3
If n = 3 then u₃ = a + b(3) + c(3)² = 12
= a + 3b + 9c = 12

Each value of u_n is a multiple of 3, hence the answers (a, b and c) are also multiples of 3, so I’m guessing that I need to use some kind of ratio solution between each new quadratic to find the answer. The difficulty I’m having is that other quadratic equations I’ve solved for already had values for the coefficients, so I’m uncertain on what quadratic formula I need at this point. I probably don’t need one, but I’m definitely stuck either way.

 

[micromass]

Hi odolwa99!

There is no need for quadratic formula's. You have a system of three equations and three unknowns:

a + b + c = 0
a + 2b + 4c = 3
a + 3b + 9c = 12

Solve it.

 

[odolwa99]

Ok, thanks for the tip. I'll give it a second look.