Suppose \( \$ 3,900 \) is deposited into an account which has an annual percentage rate of \( 7.77 \% \) per year. Assume it remains in the account for 22 years and no additional money is added to the account other than interest. (a) Complete the boxes below to write an expression for the amount (in dollars) at year \( x=22 \), of assuming interest is compounded annually. Do not round any values. You can enter arithmetic expressions (containing \( +,-,{ }^{*}, / \), or \( \wedge \) ) in any of these boxes. 22 3900 - ( \( 1+0.0777 \) ) What is the value in year \( x=22 \) of an investment of \( \$ 3,900 \) dollars which pays \( 7.77 \% \) compounded annually? \$ \( \square \) 20,231.03 (Round to the nearest 0.01 dollars)

When you're looking at compounded interest, you're essentially putting your money on a magic carpet ride – one that goes up! In this case, the formula for compound interest is given by \( A = P(1 + r)^t \), where \( P \) is the principal amount (your initial deposit), \( r \) is the annual interest rate (in decimal), and \( t \) is the number of years. So, your expression at year \( x = 22 \) would be \( 3900 \times (1 + 0.0777)^{22} \). If you plug in the numbers, \( 3900 \times (1.0777)^{22} \) will take you on that fantastic ride to the future value! After all the calculations, you'll find that your original deposit of \( \$3900 \) grows to about \( \$20,231.03 \) in 22 years. Keep in mind, that’s the magic of compounding interest – patience truly pays off!