# Math Help - 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 $x^3-2x^2-x+2$ whose roots are -1, 1 and 2

$a_n = \alpha \-1)^n + \beta \: 1^n + \gamma \: 2^n" alt="a_n = \alpha \-1)^n + \beta \: 1^n + \gamma \: 2^n" />

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

$a_n = -2 \-1)^n + 6 \:\: 1^n - 2^n" alt="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
$A_n - A_{n-1} = -n$

$....$

$A_1 - A_0 = -1$

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

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