$$y_n=\rho^ny_0+(1+\rho+\rho^2+\cdots+\rho^{n-1})b.$$ If $\rho \not=1$, we can write this solution in the more compact form $$y_n=\rho^ny_0+\frac{1-\rho^n}{1-\rho}b.$$
This is from Elem. Diff. Eq. - Boyce, DiPrima.
How was the more compact form derived? In my calculus text: $1+p+p^2+\dotsb+p^{n-1}=\frac1{1-p}$.
So where did Boyce/DiPrima get that numerator from?