You want to be able to withdraw $$ \$ 35,000 $$ from your account each year for 25 years after you retire. You expect to retire in 20 years. If your account earns $$ 10 \% $$ interest, how much will you need to deposit each year until retirement to achieve your retirement goals? $$ \$ $$ Round your answer to the nearest cent.

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Let the amount needed at retirement be $$ A $$. Since you wish to withdraw $$\$35,\!000$$ per year for 25 years at an annual interest rate of $$10\%$$, the present value of this annuity at retirement is given by

$$
A = 35000 \times \frac{1 - (1.10)^{-25}}{0.10}.
$$

**Step 1.** Compute $$A$$:

$$
(1.10)^{-25} = \frac{1}{(1.10)^{25}}.
$$

A typical calculation yields $$(1.10)^{25} \approx 10.8333$$, so

$$
(1.10)^{-25} \approx \frac{1}{10.8333} \approx 0.0923.
$$

Thus,

$$
A \approx 35000 \times \frac{1 - 0.0923}{0.10} = 35000 \times \frac{0.9077}{0.10} \approx 35000 \times 9.077 = 317695.
$$

So, you need approximately $$\$317,\!695$$ at retirement.

**Step 2.** Find the annual deposit $$X$$ required. You plan to make annual deposits at the end of each year for 20 years, and the future value of an annuity (with interest rate $$10\%$$) is

$$
\text{Future Value} = X \times \frac{(1.10)^{20} - 1}{0.10}.
$$

We set this equal to $$A$$:

$$
X \times \frac{(1.10)^{20} - 1}{0.10} = 317695.
$$

**Step 3.** Solve for $$X$$:

$$
X = \frac{317695 \times 0.10}{(1.10)^{20} - 1}.
$$

A calculation gives $$(1.10)^{20} \approx 6.7275$$, so

$$
(1.10)^{20} - 1 \approx 6.7275 - 1 = 5.7275.
$$

Thus,

$$
X \approx \frac{31769.5}{5.7275} \approx 5549.24.
$$

**Final Answer:**

You need to deposit approximately $$\$5549.24$$ each year for 20 years to achieve your retirement goal.
You need to deposit approximately \$5,549.24 each year until retirement to achieve your retirement goals.

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