The value of a collector's item is expected to increase exponentially each year. The item is purchased for \( \$ 500 \) and is value increases at a rate of \( 5 \% \) per year. Find the value of the item after 4 years. \( \$ 578.81 \) \( \$ 607.75 \) \( \$ 1687.50 \) \( \$ 2531.25 \)

To find the value of the item after 4 years with an initial price of \( \$ 500 \) and an annual increase of \( 5\% \), we can use the formula for exponential growth, which is: \[ V = P (1 + r)^t \] Where: - \( V \) is the future value of the investment/loan, including interest. - \( P \) is the principal investment amount (the initial deposit). - \( r \) is the annual interest rate (decimal). - \( t \) is the number of years the money is invested or borrowed for. Plugging in the numbers: \[ V = 500 (1 + 0.05)^4 \] \[ V = 500 (1.21550625) \] \[ V \approx 607.75 \] So, the value of the item after 4 years will be approximately \( \$607.75 \). Thus, the correct answer is: \( \$ 607.75 \). If you were to hold on to your collectible longer, imagine the value soaring even higher! The real beauty of exponential growth is that it’s like a snowball effect, gaining momentum over time. Just like those rare comic books or vintage vinyl records that get more precious as years go by! Interestingly, many collectors focus not only on their love for the items but also on market trends. Understanding what drives value in collectibles can help you make savvy investment decisions. Researching auction results, joining collector forums, and attending conventions can all provide you with valuable insights into which items to hold onto and which to potentially let go. Happy collecting!