difference equation

**manalive04** · February 5th 2009, 10:58 AM

I am very stuck as to how to attempt this question, would really appreciate some help as lecturer was not clear

xn+6 - 3xn+4 + 3xn+2 - xn = 0

given initial condition

X0=X2=X4=0, X1=2, X3=12, X5=30

Please help me, would be so grateful

**Soroban** · February 5th 2009, 12:21 PM

Hello, manalive04!

I have an "eyeball" solution . . .

$x_{n+6} - 3x_{n+4} + 3x_{n+2} - x_n \;=\;0$

Given initial conditions:

. . $\begin{array}{c}x_0\:=\:0\\<br /> x_1 \:=\:2 \\ x_2 \:=\:0 \\ x_3 \:=\:12 \\ x_4 \:=\:0 \\ x_5 \:=\:30 \\<br /> \vdots \end{array}$

We have: . $\begin{array}{ccc}x_0 &=&0 \\ x_1 &=&1\!\cdot\!2 \\ x_2 &=& 0 \\ x_3 &=&3\!\cdot\!4 \\ x_4&=& 0 \\ x_5 &=& 5\!\cdot\!6 \\ \vdots & &\vdots \end{array}$

We see that: . $x_n \;=\;\begin{Bmatrix}0 & \text{for even }n \\ n(n+1) & \text{ for odd }n \end{Bmatrix}$

Therefore, the general term is: . $x_n \;=\;\frac{1 - (-1)^n}{2}\,n(n+1)$

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