Let’s say you plan to invest $200 at the start of each year into an account that averages 3% interest compounded annually at the end of the year. How many years until the account has more than $10,000? $20,000?

We know that, at the end of year 1, the amount in the account is $206. At the end of year 2, the amount in the account is $418.18 since . At the start of year 30, for example, that original $200 has been compounded a total of 29 times, while the last $200 deposited has been compounded 0 times. Figuring out how much is in the account 30 years after the first deposit means adding up . We can use the formula for the sum of a geometric sequence, , to find the total amount in the account. The sequence starts at and increases at a rate of each year. After years, the total in the account is . Now we have a simpler expression to evaluate for different -values. It turns out that when , the account has about $10,301 in it, and when , it has about $20,682 in it.