A couple deposits $$ \$ 21,000 $$ into an account earning $$ 6 \% $$ annual interest for 15 years. Calculate the future value of the investment if the interest is compounded weekly. Round your answer to the nearest cent. Formulas Answer How to enter your answer (opens in new window) Keyboard Shortcuts

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To calculate the future value of an investment with compound interest, we can use the formula:

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

Where:
- $$ FV $$ = future value of the investment
- $$ P $$ = principal amount (initial deposit)
- $$ r $$ = annual interest rate (as a decimal)
- $$ n $$ = number of times interest is compounded per year
- $$ t $$ = number of years the money is invested or borrowed

### Given:
- $$ P = 21000 $$
- $$ r = 0.06 $$ (6% as a decimal)
- $$ n = 52 $$ (since interest is compounded weekly)
- $$ t = 15 $$

### Step-by-step Calculation:

1. **Calculate $$ \frac{r}{n} $$**:
   $$
   \frac{r}{n} = \frac{0.06}{52}
   $$

2. **Calculate $$ nt $$**:
   $$
   nt = 52 \times 15
   $$

3. **Substitute values into the future value formula**:
   $$
   FV = 21000 \left(1 + \frac{0.06}{52}\right)^{52 \times 15}
   $$

Now, let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$52\times 15$$
- step1: Multiply the numbers:
  $$780$$
Calculate or simplify the expression $$ 0.06/52 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.06}{52}$$
- step1: Convert the expressions:
  $$\frac{\frac{3}{50}}{52}$$
- step2: Multiply by the reciprocal:
  $$\frac{3}{50}\times \frac{1}{52}$$
- step3: Multiply the fractions:
  $$\frac{3}{50\times 52}$$
- step4: Multiply:
  $$\frac{3}{2600}$$
Calculate or simplify the expression $$ 21000*(1+(0.06/52))^(52*15) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$21000\left(1+\left(\frac{0.06}{52}\right)\right)^{52\times 15}$$
- step1: Divide the terms:
  $$21000\left(1+\frac{3}{2600}\right)^{52\times 15}$$
- step2: Add the numbers:
  $$21000\left(\frac{2603}{2600}\right)^{52\times 15}$$
- step3: Multiply the numbers:
  $$21000\left(\frac{2603}{2600}\right)^{780}$$
- step4: Simplify:
  $$21000\times \frac{2603^{780}}{2600^{780}}$$
- step5: Rewrite the expression:
  $$200\times 105\times \frac{2603^{780}}{2600^{780}}$$
- step6: Rewrite the expression:
  $$200\times 105\times \frac{2603^{780}}{200^{780}\times 13^{780}}$$
- step7: Reduce the numbers:
  $$105\times \frac{2603^{780}}{200^{779}\times 13^{780}}$$
- step8: Rewrite the expression:
  $$5\times 21\times \frac{2603^{780}}{200^{779}\times 13^{780}}$$
- step9: Rewrite the expression:
  $$5\times 21\times \frac{2603^{780}}{25^{779}\times 8^{779}\times 13^{780}}$$
- step10: Rewrite the expression:
  $$5\times 21\times \frac{2603^{780}}{5^{1558}\times 8^{779}\times 13^{780}}$$
- step11: Reduce the numbers:
  $$21\times \frac{2603^{780}}{5^{1557}\times 8^{779}\times 13^{780}}$$
- step12: Multiply the fractions:
  $$\frac{21\times 2603^{780}}{5^{1557}\times 8^{779}\times 13^{780}}$$
### Intermediate Calculations:

1. **Calculate $$ \frac{r}{n} $$**:
   $$
   \frac{0.06}{52} \approx 0.0011538461538461538
   $$

2. **Calculate $$ nt $$**:
   $$
   52 \times 15 = 780
   $$

3. **Substituting into the future value formula**:
   $$
   FV = 21000 \left(1 + 0.0011538461538461538\right)^{780}
   $$

4. **Calculating the future value**:
   $$
   FV \approx 21000 \times (1.0011538461538461538)^{780} \approx 51624.87
   $$

### Final Answer:
The future value of the investment, rounded to the nearest cent, is:

$$
\boxed{51624.87}
$$
The future value of the investment is \$51,624.87.

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