You want to be able to withdraw $$ \$ 20,000 $$ each year for 15 years. Your account earns $$ 4 \% $$ interest. How much do you need in your account at the beginning?

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Given that you wish to withdraw $$ \$20,\!000 $$ each year for 15 years and your account earns $$4\%$$ interest per year, we need to determine the present value (PV) of this annuity. The formula for the present value of an annuity is

$$
\text{PV} = P \times \frac{1 - (1+r)^{-n}}{r}
$$

where  
$$P$$ is the annual payment,  
$$r$$ is the annual interest rate (in decimal form), and  
$$n$$ is the number of years.

**Step 1. Identify the variables:**  
- $$P = 20,\!000$$  
- $$r = 0.04$$  
- $$n = 15$$

**Step 2. Plug the values into the formula:**

$$
\text{PV} = 20,\!000 \times \frac{1 - (1+0.04)^{-15}}{0.04}
$$

**Step 3. Simplify the expression:**

First, calculate $$1+0.04$$:

$$
1 + 0.04 = 1.04
$$

Next, raise $$1.04$$ to the power of $$-15$$:

$$
(1.04)^{-15} = \frac{1}{(1.04)^{15}}
$$

**Step 4. Calculate $$(1.04)^{15}$$:**

Using a calculator or logarithms, we find

$$
(1.04)^{15} \approx 1.80094
$$

Thus,

$$
(1.04)^{-15} \approx \frac{1}{1.80094} \approx 0.5551
$$

**Step 5. Compute the numerator of the annuity factor:**

$$
1 - (1.04)^{-15} \approx 1 - 0.5551 = 0.4449
$$

**Step 6. Divide by the interest rate $$r$$:**

$$
\frac{0.4449}{0.04} \approx 11.1225
$$

**Step 7. Multiply by the annual payment $$P$$:**

$$
\text{PV} \approx 20,\!000 \times 11.1225 \approx 222,\!450
$$

Thus, you need approximately $$\$222,\!450$$ in the account at the beginning.
You need approximately \$222,450 in your account at the beginning.

You want to be able to withdraw each year for 15 years. Your account earns
interest. How much do you need in your account at the beginning?

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You want to be able to withdraw each year for 15 years. Your account earns interest. How much do you need in your account at the beginning?