Question Meg invested $$ \$ 16,000 $$ in a savings account. If the interest rate is $$ 6 \% $$ per year, how much will be in the account in 5 years with monthly compounding? Round your answer to the nearest cent. Do NOT round until you calculate the final answer

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To calculate the amount in the savings account after 5 years with monthly compounding, 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:
- $$ P = 16000 $$
- $$ r = 0.06 $$ (6% as a decimal)
- $$ n = 12 $$ (monthly compounding)
- $$ t = 5 $$

Now, we can substitute these values into the formula:

$$
A = 16000 \left(1 + \frac{0.06}{12}\right)^{12 \times 5}
$$

Let's calculate this step by step.

1. Calculate $$ \frac{r}{n} $$:
   $$
   \frac{0.06}{12} = 0.005
   $$

2. Calculate $$ nt $$:
   $$
   12 \times 5 = 60
   $$

3. Calculate $$ A $$:
   $$
   A = 16000 \left(1 + 0.005\right)^{60}
   $$

4. Calculate $$ 1 + 0.005 $$:
   $$
   1 + 0.005 = 1.005
   $$

5. Now calculate $$ (1.005)^{60} $$:
   $$
   (1.005)^{60}
   $$

6. Finally, multiply by $$ 16000 $$.

Let's perform the calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$16000\left(1+\frac{0.06}{12}\right)^{12\times 5}$$
- step1: Divide the terms:
  $$16000\left(1+\frac{1}{200}\right)^{12\times 5}$$
- step2: Add the numbers:
  $$16000\left(\frac{201}{200}\right)^{12\times 5}$$
- step3: Multiply the numbers:
  $$16000\left(\frac{201}{200}\right)^{60}$$
- step4: Simplify:
  $$16000\times \frac{201^{60}}{200^{60}}$$
- step5: Rewrite the expression:
  $$200\times 80\times \frac{201^{60}}{200^{60}}$$
- step6: Reduce the numbers:
  $$80\times \frac{201^{60}}{200^{59}}$$
- step7: Rewrite the expression:
  $$40\times 2\times \frac{201^{60}}{200^{59}}$$
- step8: Rewrite the expression:
  $$40\times 2\times \frac{201^{60}}{40^{59}\times 5^{59}}$$
- step9: Reduce the numbers:
  $$2\times \frac{201^{60}}{40^{58}\times 5^{59}}$$
- step10: Rewrite the expression:
  $$2\times \frac{201^{60}}{8^{58}\times 5^{58}\times 5^{59}}$$
- step11: Rewrite the expression:
  $$2\times \frac{201^{60}}{2^{174}\times 5^{58}\times 5^{59}}$$
- step12: Reduce the numbers:
  $$1\times \frac{201^{60}}{2^{173}\times 5^{58}\times 5^{59}}$$
- step13: Multiply the fractions:
  $$\frac{201^{60}}{2^{173}\times 5^{117}}$$
After performing the calculations, we find that the amount in the account after 5 years with monthly compounding is approximately:

$$
A \approx 21581.602441
$$

Rounding this to the nearest cent, the final amount in the account will be:

$$
A \approx 21581.60
$$

Thus, the total amount in the account after 5 years will be **\$21,581.60**.
After 5 years with monthly compounding at a 6% interest rate, Meg will have approximately \$21,581.60 in her savings account.

Question Meg invested \( \$ 16,000 \) in a savings account. If the interest rate is \( 6 \% \) per year, how much will be in the account in 5 years with monthly compounding? Round your answer to the nearest cent. Do NOT round until you calculate the final answer

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