Obtain the sol'n for an=5an-2 - 4an-4

With:

a0 = 3

a1 = 2

a2 = 6

a3 = 8

Note:

The 5an-2 minus 4an-4 actually appear in the the problem as 5a and 4a with the terms being n-2 and n-4 list slightly lower than the an term

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- Aug 9th 2009, 08:14 AMorendaclSolve the equation
Obtain the sol'n for an=5an-2 - 4an-4

With:

a0 = 3

a1 = 2

a2 = 6

a3 = 8

Note:

The 5an-2 minus 4an-4 actually appear in the the problem as 5a and 4a with the terms being n-2 and n-4 list slightly lower than the an term - Aug 9th 2009, 08:35 AMred_dog
$\displaystyle a_n-5a_{n-2}+4a_{n-4}=0$

The characteristic equation is $\displaystyle t^4-5t^2+4=0$ with the roots

$\displaystyle t_1=1, \ t_2=-1, \ t_3=2, \ t_4=-2$

We have to find the general term $\displaystyle a_n$ in the form $\displaystyle a_n=a+b(-1)^n+c\cdot 2^n+d(-2)^n$

For $\displaystyle n=0, \ 1, \ 2, \ 3$ we have

$\displaystyle a_0=a+b+c+d=3$

$\displaystyle a_1=a-b+2c-2d=2$

$\displaystyle a_2=a+b+4c+4d=6$

$\displaystyle a_3=a-b+8c-8d=8$

Now find a, b, c, d.