Simple Affine Map Q

I dunno if I'm being really stupid, but I can't see how to get a solution given.

Consider the affine map given by:

x_{n+1} = a + bx_{n}

Show the solution is given by:

x_n = b^n + \frac{a}{1-b}

The solution says to just make sure it satisfies the affine map, with the first line as:

x_{n+1} = (x_0 - \frac{a}{1-b})b^{n+1} + \frac{a}{1-b}

I can't see where that came from. Thanks!

Reply 1

TheJ0ker

Original post by Brit_Miller

You could treat it as a second order difference equation and solve it I think,

i.e solve $x_{n+1}-bx_n + 0x_{n-1} = a$ for $x_n$

Wait just sub the solution in and show it satisfies the equation

if

x_n = b^n + \frac{a}{1-b}

then what does

x_{n+1}=

? (just replace the ns for n+1s)

(edited 11 years ago)

Reply 2

Brit_Miller

Original post by TheJ0ker

You could treat it as a second order difference equation and solve it I think,

i.e solve

x_{n+1}-bx_n + 0x_{n-1} = a

for

x_n

Wait just sub the solution in and show it satisfies the equation

I thought I was going down the wrong road with that as I just got

x_{n+1} = b^{n+1} + \frac{a}{1-b}

I wasn't sure where to go from there.

Reply 3