I just posted this
http://www.mathhelpforum.com/math-he...-sequence.html
- it might help.
Added note. In the case of repeated roots of the characteristic equation, solutions are of the form
$\displaystyle a_n = \left( c_1 n + c_2 \right) \rho^n$
Let's try the first one. Substitute $\displaystyle a_n = c \rho^n$ into
$\displaystyle a_n - 7 a_{n-1} + 10 a_{n-2} = 0$
gives
$\displaystyle c \rho^n - 7 c \rho^{n-1} + 10 c \rho^{n-2} = 0$
or
$\displaystyle (\rho^2 - 7 \rho + 10) c \rho^{n-2} = 0$
which gives
$\displaystyle (\rho^2 - 7 \rho + 10) = 0$ or $\displaystyle (\rho - 2) \rho - 5) = 0$ so $\displaystyle \rho = 2,\; 5$
So, the general solution to the difference equation is
$\displaystyle a_n = c_1 2 ^n + c_2 5^n$
Now the initial condition
$\displaystyle a_0 = c_1 + c_2 = -1$
$\displaystyle a_1 = c_1 2 + c_2 5 = 4$
Two equations for $\displaystyle c_1\; \text{and}\; c_2$. SOlving gives
$\displaystyle c_1 = -3,\;\;\;c_2 = 2$
leading to the solution
$\displaystyle a_n = -3 \cdot 2 ^n + 2 \cdot 5^n$