(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. What is the value in year \( x=22 \) of an investment of \( \$ 3,900 \) dollars which pays \( 7.77 \% \) compounded annually? \( \$ 20,231.03 \) Nound to the nearest 0.01 dollars)

To find the amount of an investment with compounded interest, you can use the formula for compound interest: \[ A = P(1 + r)^n \] where \( A \) is the amount of money accumulated after n years, including interest, \( P \) is the principal amount (the initial investment), \( r \) is the annual interest rate (decimal), and \( n \) is the number of years the money is invested or borrowed. In this case, you have \( P = 3900 \), \( r = 0.0777 \), and \( n = 22 \). Plugging these values into the formula gives: \[ A = 3900(1 + 0.0777)^{22} \] Now, calculate \( (1 + 0.0777)^{22} \): 1. First, calculate \( 1 + 0.0777 = 1.0777 \). 2. Then raise \( 1.0777 \) to the power of \( 22 \). After computing \( (1.0777)^{22} \), you would multiply the result by \( 3900 \) to find the total amount \( A \) in year \( x = 22 \). This calculation leads to a final value of approximately \( \$ 20,231.03 \), which matches the prompt. Let’s keep the calculator handy because compounding interest can add up quickly, and who doesn’t love watching that number grow! Remember, every year matters, so time really is your best friend when investing!