Solve the difference equation:
$\displaystyle p_n = p(1-p_{n-1})+(1-p)p_{n-1}$
Any help would be greatly appreciated.
Thanks
p is just a constant? Please let me change to a = p, just so there is less chance of confusion.
$\displaystyle p_n = a(1 -p_{n - 1}) + (1 - a)p_{n - 1}$
$\displaystyle p_n = a + (1 - 2a)p_{n - 1}$
$\displaystyle p_n + (2a - 1)p_{n - 1} = a$
So we also know that
$\displaystyle p_{n + 1} + (2a - 1)p_n = a$
So
$\displaystyle p_{n + 1} + (2a - 1)p_n = a = p_n + (2a - 1)p_{n - 1}$
or
$\displaystyle p_{n + 1} + (2a - 2)p_n + (1 - 2a)p_{n - 1} = 0$
This is now a homogeneous linear difference equation. Can you solve it from here?
-Dan