# Quadratic Recurrence Relation

1. Jul 17, 2011

### odolwa99

1. The problem statement, all variables and given/known data

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

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

2. Relevant equations

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

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 un 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.

2. Jul 17, 2011

### 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.

3. Jul 17, 2011

### odolwa99

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

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