How much would you have to deposit in an account with a \( 5 \% \) interest rate, compounded annually, to have \( \$ 1300 \) in your account 13 years later? \[ P=\$[?] \] \( F=P\left(1+\frac{r}{n}\right)^{n t} \) Round to the nearest cent.

To find out how much you would need to deposit now to reach \( \$1300 \) in 13 years at a \( 5\% \) interest rate compounded annually, we can rearrange the formula for compound interest. Here, \( F \) is the future value (\$1300), \( r \) is the interest rate (0.05), \( n \) is the number of times interest is compounded per year (1), and \( t \) is the time in years (13). So, we set up the equation: \[ P = \frac{F}{\left(1 + \frac{r}{n}\right)^{n t}} = \frac{1300}{\left(1 + \frac{0.05}{1}\right)^{1 \times 13}} = \frac{1300}{(1.05)^{13}} \] Now, we calculate: \[ (1.05)^{13} \approx 1.7137 \] \[ P \approx \frac{1300}{1.7137} \approx 758.68 \] Therefore, you would need to deposit approximately \( P = \$758.68 \). Now, for a little extra insight, did you know that the concept of compound interest has been around for centuries? The earliest recorded use can be traced back to the ancient civilizations of Mesopotamia! Not only does it highlight the power of saving and investing over time, but understanding such historical contexts can give you a richer appreciation of finance as a whole. If you're ever feeling bogged down by numbers and calculations, remember to always break complex problems into smaller parts! Instead of diving into the formula blindly, establish what each variable represents, then tackle calculations step by step. This approach not only helps prevent mistakes but also builds your confidence in tackling similar problems in the future!