Each year for $10$ years, the population of a city increased by $5%$ of its value in the previous year. If the initial population is $200,000$, what was the population after $10$ years?
My solution: $$ (200000)(1.05)^{n-1} (200000)(1.05)^{10-1} (200000)(1.05)^{9} $$ I see on the internet for the same question, people are getting $(200000)(1.05)^{10}$ instead. Why is this so? I drew "$10$ lines", and started counting starting from the first line and got $9$, so therefore the population increases by $5\%$ $9$ times, not $10$. Could anyone clear this up?
Thanks!