# Thread: Find solution to a recurrence relation...

1. ## Find solution to a recurrence relation...

I am stuck on this question, and do not know where to head with it.

Find the solution to An= 2An-1 + An-2 - 2An-3 for n=2,3,4,5......, with a0=3, a1=6 and a2=0

(Note that n-1, n-2, n-3 are subscripts)

2. Hi

Well it depends by your level in maths
The characteristic polynomial is $\displaystyle x^3-2x^2-x+2$ whose roots are -1, 1 and 2

$\displaystyle a_n = \alpha \-1)^n + \beta \: 1^n + \gamma \: 2^n$

$\displaystyle \alpha$, $\displaystyle \beta$, $\displaystyle \gamma$ values are found by the values of $\displaystyle a_0$, $\displaystyle a_1$, $\displaystyle a_2$

$\displaystyle a_n = -2 \-1)^n + 6 \:\: 1^n - 2^n$

3. Thank you

Would I approach An = An-1 -n, a0=4 the same way?

4. No
$\displaystyle A_n - A_{n-1} = -n$

$\displaystyle ....$

$\displaystyle A_1 - A_0 = -1$

By summation
$\displaystyle A_n - A_0 = -\sum_{k=1}^n\:k = -\frac{n(n+1)}{2}$

$\displaystyle A_n = 4-\frac{n(n+1)}{2}$