Equation in finite differences

**Angus** · May 12th 2009, 09:45 AM

Hi all!

Can someone help me with this exercise? I don't know how to start

Calculate the equation in finite differences of:

14x[n]-9x[n-1]+x[n-2]=70delta[n+3]-10delta[n+2]

Thanks in advance
Greetings

**Media_Man** · May 22nd 2009, 09:35 AM

First, realize that $\Delta_n=x_{n+1}-x_n$ by definition. So, substitute:

$14x_n-9x_{n-1}+x_{n-2}=70(x_{n+4}-x_{n+3})-10(x_{n+2}-x_{n+1})$

The way recursive formulas are defined is by relating $x_n$ to a formula involving several values $x_k$ for $k<n$ , so we want to isolate the x with the highest subscript, in this case, $x_{n+4}$ .

$x_{n+4}=\frac{1}{70}(80x_{n+3}-10x_{n+2}+0x_{n+1}+14x_n-9x_{n-1}+x_{n-2})$

More formally,

$x_{n+6}=\frac{1}{70}(80x_{n+5}-10x_{n+4}+0x_{n+3}+14x_{n+2}-9x_{n+1}+x_n)$ for all $n\in\mathbb{N}$

Therefore, to actually find the terms in this sequence, we would need to know the first six values of the sequence.

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