# Finding fixed numbers

• Mar 8th 2010, 08:57 PM
tigergirl
Finding fixed numbers
For a fixed number a define $\Delta_{a} x_{n} = x_{n+1} - ax_n$. Find formula for $\Delta^{2}_{a}x_{n}$ and $\Delta^{3}_{a}x_{n}$. Express $\Delta_{a}x_{n}$ in terms of $\Delta x_n$ and $x_n$.

Am I suppose to find the derivative??? I'm confused.
• Mar 8th 2010, 10:47 PM
chisigma
From the definition...

$\Delta_{a} x_{n} = x_{n+1} - a\cdot x_{n}$ (1)

... follows...

$\Delta x_{n} = x_{n+1} - x_{n} = \Delta_{a}x_{n} + (a-1)\cdot x_{n}$ (2)

$\Delta_{a}^{2} x_{n}$ and $\Delta_{a}^{3} x_{n}$ can also be derived from (1)

Kind regards

$\chi$ $\sigma$