This is a question from the book Discrete Mathematical Structures by Bernard Kolman, Robert C. Busby and Sharon Cutler Ross.
I want to find the explicit formula of the recurrence relation $g_n = 2g_{n-1} - 2g_{n-2}$ with initial conditions $g_1 = 1$ and $g_2 = 4$.
I form the characteristic equation $x^2 = 2x - 2$. I get $1+i$ and $1-i$ as roots.
Since there are 2 distinct roots, $g_n = us_1^n + vs_2^n$.
$g_1 = u(1+i)^1 + v(1-i)^1$
$g_2 = u(1+i)^2 + v(1-i)^2$
Simplyfing:
$(1+i)u + (1-i)v = 1$
$(2i)u + (-2i)v = 4$
How do I solve this?
Dividing your second equation by $2i$ and rearrange you get $$u-v=-2i$$ Inserting the expression for $u$ into the first and simplifying (try it!) you get $$2v-2i+1=0 \leftrightarrow v=i-\frac{1}{2}$$