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I'm having a lot of trouble understanding second-order linear homogenous recurrence relations. The notes I have been given really are not helping me understand the process as they are written for mathematics students (which I am not) so there is a lot of assumed knowledge that is just left out. I was wondering if someone could explain how I would solve this problem step by step

$a_n = −4a_{n−1} − 4a_{n−2}$ for all integers $n ≥ 2$ with $a_0 = 0$, and $a_1 = −1$

Tortar
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hunter
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1 Answers1

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Solving linear recurrence equations is similar to solving linear differential equations. $$a_n = −4a_{n−1} − 4a_{n−2}$$

$$ a_n= b^n \implies b^2+4b+4=0$$ $$b=-2,-2$$

Thus the solution is $$a_n = A (-2)^n + B n(-2)^n $$

The coefficients A and B are found from the initial conditions. For $n=0$, we get $A=0$ and for $n=1$ we get $-2B=-1$, thus $B=1/2$

The solution is $$ a_n = (-1)^n n 2^{n-1}$$

Mohammad Riazi-Kermani
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