Use the formula for present value of money to calculate the amount you need to invest now in one lump sum in order to have $$ \$ 100,000 $$ after 18 years with an ApR of $$ 9 \% $$ compounded quarterly Round your answer to the nearest cent. If necessary.

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To calculate the present value (PV) of money needed to achieve a future value (FV) of $$ \$100,000 $$ after 18 years with an annual percentage rate (APR) of $$ 9\% $$ compounded quarterly, we can use the present value formula for compound interest:

$$
PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}
$$

Where:
- $$ FV $$ = future value (\$100,000)
- $$ r $$ = annual interest rate (0.09)
- $$ n $$ = number of times interest is compounded per year (4 for quarterly)
- $$ t $$ = number of years (18)

Now, let's plug in the values and calculate the present value step by step.

1. Calculate $$ \frac{r}{n} $$:
   $$
   \frac{r}{n} = \frac{0.09}{4}
   $$

2. Calculate $$ nt $$:
   $$
   nt = 4 \times 18
   $$

3. Calculate $$ (1 + \frac{r}{n})^{nt} $$:
   $$
   (1 + \frac{r}{n})^{nt}
   $$

4. Finally, calculate $$ PV $$:
   $$
   PV = \frac{100000}{(1 + \frac{r}{n})^{nt}}
   $$

Let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$4\times 18$$
- step1: Multiply the numbers:
  $$72$$
Calculate or simplify the expression $$ 0.09/4 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.09}{4}$$
- step1: Convert the expressions:
  $$\frac{\frac{9}{100}}{4}$$
- step2: Multiply by the reciprocal:
  $$\frac{9}{100}\times \frac{1}{4}$$
- step3: Multiply the fractions:
  $$\frac{9}{100\times 4}$$
- step4: Multiply:
  $$\frac{9}{400}$$
Calculate or simplify the expression $$ (1 + 0.0225)^72 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(1+0.0225\right)^{72}$$
- step1: Add the numbers:
  $$1.0225^{72}$$
- step2: Convert the expressions:
  $$\left(\frac{409}{400}\right)^{72}$$
- step3: Use the properties of exponents:
  $$\frac{409^{72}}{400^{72}}$$
Calculate or simplify the expression $$ 100000/(4.963166) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{100000}{4.963166}$$
- step1: Convert the expressions:
  $$\frac{100000}{\frac{2481583}{500000}}$$
- step2: Multiply by the reciprocal:
  $$100000\times \frac{500000}{2481583}$$
- step3: Multiply:
  $$\frac{100000\times 500000}{2481583}$$
- step4: Multiply:
  $$\frac{50000000000}{2481583}$$
Now that we have completed the calculations, let's summarize the steps and results:

1. **Calculate $$ \frac{r}{n} $$**:
   $$
   \frac{r}{n} = 0.0225
   $$

2. **Calculate $$ nt $$**:
   $$
   nt = 72
   $$

3. **Calculate $$ (1 + \frac{r}{n})^{nt} $$**:
   $$
   (1 + 0.0225)^{72} \approx 4.963166
   $$

4. **Calculate the present value $$ PV $$**:
   $$
   PV = \frac{100000}{4.963166} \approx 20148.43
   $$

Thus, the amount you need to invest now in one lump sum to have $$ \$100,000 $$ after 18 years at an APR of $$ 9\% $$ compounded quarterly is approximately **\$20,148.43**.
You need to invest approximately \$20,148.43 now to have \$100,000 after 18 years at a 9% annual interest rate compounded quarterly.

Use the formula for present value of money to calculate the amount you need to invest now in one lump sum in order to have \( \$ 100,000 \) after 18 years with an ApR of \( 9 \% \) compounded quarterly Round your answer to the nearest cent. If necessary.

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