Question about a nonhomogeneous recurrence

Question

Im trying to solve this nonhomogenous recurrence: link to the nonhomogeneous recurrence

The homogeneous part is $A*(-5)^n + B*2^n$

The part that is bothering me is $7 * 5^n$ does that translate to $Cn * 5^n$ ?

Answer 1

The solutions to the associated homogeneous equation are given by $$r^2-3r-10=0$$ $$(r+2)(r-5)=0$$ $$r=-2,5$$ Hence we have the complementary function $$c_n=A(-2)^n+B(5)^n$$ Given this information the particular function will be of the form $$p_n=Cn(5)^n$$ So plugging this into the original equation, we then have $$Cn(5)^n-3C(n-1)(5)^{n-1}-10C(n-2)(5)^{n-2}=7(5)^n$$ $$\left(1-\frac35-\frac25\right)Cn(5)^n+\left(\frac35+\frac45\right)C(5)^n=7(5)^n$$ $$\frac75C(5)^n=7(5)^n$$ $$\therefore C=5$$ So the general solution is $$a_n=c_n+p_n=A(-2)^n+(B+5n)(5)^n$$