How much should you deposit at the end of each month into an investment account that pays $$ 6.5 \% $$ compounded monthly to have $$ \$ 3 $$ million when you retire in 38 years? How much of the $$ \$ 3 $$ million comes from interest? (i) Click the icon to view some finance formulas.

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We want to find the monthly deposit $$ P $$ such that after 38 years (with 12 deposits per year) the annuity grows to $$ \$3,\!000,\!000 $$ when the account earns $$ 6.5\% $$ compounded monthly. The future value of an annuity with end‐of‐period payments is given by

$$
FV = P \cdot \frac{(1+i)^n - 1}{i},
$$

where

- $$ i $$ is the periodic interest rate,
- $$ n $$ is the total number of deposits.

Since the annual interest rate is $$ 6.5\% $$ compounded monthly, the monthly interest rate is

$$
i = \frac{0.065}{12}.
$$

The total number of months is

$$
n = 38 \times 12 = 456.
$$

We need the annuity to equal $$ 3,\!000,\!000 $$, so we set

$$
P \cdot \frac{(1+i)^n - 1}{i} = 3,\!000,\!000.
$$

Solving for $$ P $$ we get

$$
P = \frac{3,\!000,\!000 \cdot i}{(1+i)^n - 1}.
$$

Substitute in $$ i = \frac{0.065}{12} $$ and $$ n = 456 $$:

$$
P = \frac{3,\!000,\!000 \cdot \frac{0.065}{12}}{\left(1+\frac{0.065}{12}\right)^{456}-1}.
$$

To understand the computation better, we perform the steps:

1. Compute the monthly interest rate:
   $$
   i = \frac{0.065}{12} \approx 0.00541667.
   $$

2. Compute the accumulation factor:
   $$
   \left(1+i\right)^{456} = \left(1.00541667\right)^{456}.
   $$
   Taking logarithms,
   $$
   \ln\left(1.00541667\right) \approx 0.00540217,
   $$
   so
   $$
   456 \cdot 0.00540217 \approx 2.462,
   $$
   and
   $$
   \left(1.00541667\right)^{456} \approx e^{2.462} \approx 11.73.
   $$

3. Then,
   $$
   (1+i)^n - 1 \approx 11.73 - 1 = 10.73.
   $$

4. Now find $$ P $$:
   $$
   P \approx \frac{3,\!000,\!000 \times 0.00541667}{10.73} \approx \frac{16,\!250}{10.73} \approx 1515.
   $$
   (The slight differences are due to rounding at each step, but the monthly deposit is approximately $$ \$1,\!515 $$.)

So, you should deposit approximately

$$
\boxed{\$1,\!515 \text{ per month}}
$$

into the account.

Next, we calculate how much of the $$ \$3,\!000,\!000 $$ comes from interest. The total amount contributed over 456 months is

$$
\text{Total Contributions} = P \cdot 456 \approx 1,\!515 \times 456 \approx 690,\!000.
$$

The interest earned is then

$$
\text{Interest Earned} = 3,\!000,\!000 - 690,\!000 \approx 2,\!310,\!000.
$$

Thus, about

$$
\boxed{\$2,\!310,\!000}
$$

of the $$ \$3,\!000,\!000 $$ comes from interest.
You should deposit approximately \$1,515 each month. About \$2,310,000 of the \$3,000,000 comes from interest.

How much should you deposit at the end of each month into an investment account that pays \( 6.5 \% \) compounded monthly to have \( \$ 3 \) million when you retire in 38 years? How much of the \( \$ 3 \) million comes from interest? (i) Click the icon to view some finance formulas.

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