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 $$ \square $$ $$ 3900 $$ - ( $$ 1+0.0777 $$ ) 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)

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To find the amount in the account after 22 years with an annual interest rate of 7.77% compounded annually, we can use the formula for compound interest:

$$ A = P \left( 1 + \frac{r}{n} \right)^{nt} $$

where:
- $$ A $$ is the amount in the account after $$ t $$ years,
- $$ P $$ is the principal amount (initial deposit),
- $$ r $$ is the annual interest rate (in decimal form),
- $$ n $$ is the number of times interest is compounded per year,
- $$ t $$ is the time in years.

Given:
- $$ P = \$3,900 $$,
- $$ r = 7.77\% = 0.0777 $$,
- $$ n = 1 $$ (compounded annually),
- $$ t = 22 $$ years.

Substitute these values into the formula:

$$ A = 3900 \left( 1 + \frac{0.0777}{1} \right)^{1 \times 22} $$

$$ A = 3900 \left( 1 + 0.0777 \right)^{22} $$

$$ A = 3900 \left( 1.0777 \right)^{22} $$

Now, we can calculate the value of $$ A $$ using a calculator or by simplifying the expression further.
Calculate the value by following steps:
- step0: Calculate:
  $$3900\left(1+0.0777\right)^{22}$$
- step1: Add the numbers:
  $$3900\times 1.0777^{22}$$
- step2: Convert the expressions:
  $$3900\left(\frac{10777}{10000}\right)^{22}$$
- step3: Simplify:
  $$3900\times \frac{10777^{22}}{10000^{22}}$$
- step4: Rewrite the expression:
  $$100\times 39\times \frac{10777^{22}}{10000^{22}}$$
- step5: Rewrite the expression:
  $$100\times 39\times \frac{10777^{22}}{100^{44}}$$
- step6: Reduce the numbers:
  $$39\times \frac{10777^{22}}{100^{43}}$$
- step7: Multiply:
  $$\frac{39\times 10777^{22}}{100^{43}}$$
The amount in the account after 22 years with an annual interest rate of 7.77% compounded annually is approximately $20,231.03 (rounded to the nearest 0.01 dollars).
After 22 years, the investment of \$3,900 at an annual interest rate of 7.77% compounded annually grows to approximately \$20,231.03.

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