I was trying to figure out how to graph a number that grows by 200 each week and the total decays by 5% and eventually found the recurrence $$g(0)=0, g(n)=200 + 0.95 g(-1 + n)$$ which reduces to $$-4000 * (-1 + (19/20)^n)$$ Sure, that works.
In trying to find this answer I accidentally found an answer in sigma notation that has me scratching my head. $$\sum_{i=1}^n 200*2.78951^{-0.05*(n-1)}$$ How does this almost perfectly reflect the recurrence and where does this 2.78951 come from?!?! I came to the number by noticing using e was very close and adjusted the number until it was almost perfect.
$(19/20)^{-1/0.05}=2.78951$ is a special case of the large-$n$ approximation $(1-1/n)^{-n}\approx e$.