Rashon is going to invest in an account paying an interest rate of \( 6.9 \% \) compounded quarterly. How much would Rashon need to invest, to the nearest dollar, for the value of the account to reach \( \$ 20,200 \) in 13 years?

To determine how much Rashon needs to invest today, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \(A\) is the amount of money accumulated after n years, including interest. - \(P\) is the principal amount (the initial amount of money). - \(r\) is the annual interest rate (decimal). - \(n\) is the number of times that interest is compounded per year. - \(t\) is the number of years the money is invested or borrowed. Given: - \(A = 20,200\) - \(r = 0.069\) (which is \(6.9\%\)) - \(n = 4\) (quarterly) - \(t = 13\) We need to solve for \(P\): \[ 20,200 = P \left(1 + \frac{0.069}{4}\right)^{4 \times 13} \] Calculating the values inside the parentheses: \[ 1 + \frac{0.069}{4} = 1 + 0.01725 = 1.01725 \] Now calculating the exponent: \[ 4 \times 13 = 52 \] So we can rewrite the equation: \[ 20,200 = P \left(1.01725^{52}\right) \] Calculating \(1.01725^{52}\): \[ 1.01725^{52} \approx 2.298471 \] Now substituting back to find \(P\): \[ 20,200 = P \times 2.298471 \] Now, solving for \(P\): \[ P = \frac{20,200}{2.298471} \approx 8,786.42 \] Therefore, Rashon needs to invest approximately \(P \approx 8,786\) dollars, rounded to the nearest dollar. So, Rashon should invest about **$8,786** today to reach his goal! --- Planting some financial seeds can pay off big in the future! Did you know that compounding interest works almost like a snowball effect? The more time your money has to grow, the larger it gets as interest adds up not just on your initial investment but on the interest already earned. If you're looking for something fun and a bit of a brain workout, check out some online compound interest calculators! They’re super handy for visualizing how different rates, times, and compounding frequencies can impact your investment. You might be surprised at what a difference a few extra years or a slightly different interest rate makes!