I have a recurrence relation for a fund that starts a 50 million, 6 % interest every year, and an outtake of 2 million/year. How to find out a solution for what funds exists after n years?
$$x_{n+1}=x_{n}*1.06-2$$ $$ x_{0}=50$$
Under are my calculations -
Assosiated homogenous relation : $$x_{n+1}=1.06x_{n}$$
Special solution : $$x_{n+1}=1.06x_{n}-2 = x_{n}^{s}=33.3334$$
General solution : $$x_{n}=C*(1.06)^n+33.3334$$
How to calculate the last unknown C? The answer in the book says its 50/3, but i really can't get my head around why its not just 50 as the $x_{0}$ is.
(Sorry in advance if i have misspelled some of the solution names or the equation part names, i've tried translating them from norwegian. Help on correct translations is appreciated.)