You want to be able to withdraw $$ \$ 35,000 $$ from your account each year for 30 years after you retire. You expect to retire in 20 years. If your account earns $$ 7 \% $$ interest, how much will you need to deposit each year until retirement to achieve your retirement goals?

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To determine how much you need to deposit each year until retirement to achieve your retirement goals, we can use the formula for the present value of an annuity:

$$ PV = \frac{PMT}{r} \left( 1 - \frac{1}{(1 + r)^n} \right) $$

Where:
- $$ PV $$ is the present value (the amount you need to deposit each year until retirement),
- $$ PMT $$ is the annual withdrawal amount (\$35,000),
- $$ r $$ is the annual interest rate (7%),
- $$ n $$ is the number of years until retirement (20 years).

Given:
- Annual withdrawal amount ($$ PMT $$) = \$35,000
- Annual interest rate ($$ r $$) = 7% = 0.07
- Number of years until retirement ($$ n $$) = 20 years

Substitute the given values into the formula to calculate the present value ($$ PV $$):

$$ PV = \frac{35000}{0.07} \left( 1 - \frac{1}{(1 + 0.07)^{20}} \right) $$

Now, we can calculate the present value ($$ PV $$) to determine how much you need to deposit each year until retirement.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{35000}{0.07}\times \left(1-\frac{1}{\left(1+0.07\right)^{20}}\right)$$
- step1: Add the numbers:
  $$\frac{35000}{0.07}\times \left(1-\frac{1}{1.07^{20}}\right)$$
- step2: Convert the expressions:
  $$\frac{35000}{0.07}\times \left(1-\frac{1}{\left(\frac{107}{100}\right)^{20}}\right)$$
- step3: Simplify:
  $$\frac{35000}{0.07}\times \left(1-\frac{100^{20}}{107^{20}}\right)$$
- step4: Subtract the numbers:
  $$\frac{35000}{0.07}\times \frac{107^{20}-100^{20}}{107^{20}}$$
- step5: Divide the numbers:
  $$500000\times \frac{107^{20}-100^{20}}{107^{20}}$$
- step6: Multiply:
  $$\frac{500000\left(107^{20}-100^{20}\right)}{107^{20}}$$
- step7: Multiply:
  $$\frac{500000\times 107^{20}-500000\times 100^{20}}{107^{20}}$$
The present value ($$ PV $$) is approximately \$370,790.50. This means that you need to deposit approximately \$370,790.50 each year until retirement to achieve your retirement goals of withdrawing \$35,000 each year for 30 years after retirement.
You need to deposit approximately \$370,790.50 each year until retirement to achieve your retirement goals.

You want to be able to withdraw \( \$ 35,000 \) from your account each year for 30 years after you retire. You expect to retire in 20 years. If your account earns \( 7 \% \) interest, how much will you need to deposit each year until retirement to achieve your retirement goals?

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