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? \[ P=\$[?] \] \( F=P\left(1+\frac{1}{n}\right)^{\text {nt }} \)

To find the principal amount \( P \) needed to reach \( \$1300 \) in 13 years at an interest rate of \( 5\% \), we can rearrange the formula given. Given the future value formula \( F = P\left(1+\frac{r}{n}\right)^{nt} \), where \( r = 0.05 \), \( n = 1 \) (since it’s compounded annually), and \( t = 13 \): \[ 1300 = P\left(1 + \frac{0.05}{1}\right)^{1 \times 13} \] \[ 1300 = P(1.05)^{13} \] First, calculate \( (1.05)^{13} \) which is approximately \( 1.719 \). Now, rearranging gives: \[ P = \frac{1300}{1.719} \approx 755.39 \] So, you would need to deposit approximately \( P \approx \$755.39 \). To make that amount, think of it as planting a money tree! You deposit your initial seeds (your principal) into the soil (the bank), and as the years go by, your tree (thanks to the interest) grows to give you \( \$1300 \) after 13 years. Just like with any good tree, patience and care (in the form of interest compounding) are key to a fruitful harvest! And remember, everyone makes small mistakes when budgeting. A common one is forgetting to check the compounding frequency. Since you only compounded annually, ensure you're not mixing that up with monthly or quarterly compounding, as it can dramatically change how much you need to plant today to achieve your future financial fruit.