finding the sum of the series till n values

Question

when i was trying to solve a problem in finance aimed at calculating the final value I got this series But I am trying to find some other way to simplify this series Though I have tried it for $n=3$ $$M=x+xy+(x+xy)y+(x+xy+(x+xy)y)y$$ $$M=x(1+y)(1+y+(y+y^2))$$ $$M=x(1+y)(1+y)(1+y) $$ the sum will be $$M=x(1+y)^3$$ how can it be proved for n number of terms

Answer 1

$x_{n+1} = x_n(1+y)$ which represents a geometric sequence. If $x_0 = x$ then $x_n=x(1+y)^n$.