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
First, realize that $\displaystyle \Delta_n=x_{n+1}-x_n$ by definition. So, substitute:
$\displaystyle 14x_n-9x_{n-1}+x_{n-2}=70(x_{n+4}-x_{n+3})-10(x_{n+2}-x_{n+1})$
The way recursive formulas are defined is by relating $\displaystyle x_n$ to a formula involving several values $\displaystyle x_k$ for $\displaystyle k<n$, so we want to isolate the x with the highest subscript, in this case, $\displaystyle x_{n+4}$.
$\displaystyle x_{n+4}=\frac{1}{70}(80x_{n+3}-10x_{n+2}+0x_{n+1}+14x_n-9x_{n-1}+x_{n-2})$
More formally,
$\displaystyle x_{n+6}=\frac{1}{70}(80x_{n+5}-10x_{n+4}+0x_{n+3}+14x_{n+2}-9x_{n+1}+x_n)$ for all $\displaystyle n\in\mathbb{N}$
Therefore, to actually find the terms in this sequence, we would need to know the first six values of the sequence.