To determine the amount you will have in the account after 30 years with monthly deposits and compound interest, you can use the future value formula for an annuity. The formula to calculate the future value of a series of regular deposits in an account with compound interest is:
[tex]\[ FV = P \times \left( \frac{(1 + r/n)^{nt} - 1}{r/n} \right) \][/tex]
Where:
- [tex]\( FV \)[/tex] is the future value of the account.
- [tex]\( P \)[/tex] is the amount of each deposit.
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal).
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year.
- [tex]\( t \)[/tex] is the number of years.
Given:
- [tex]\( P = 400 \)[/tex] (monthly deposit)
- [tex]\( r = 0.04 \)[/tex] (annual interest rate)
- [tex]\( n = 12 \)[/tex] (monthly compounding)
- [tex]\( t = 30 \)[/tex] (years)
Let's break down the calculation step-by-step:
1. Calculate the monthly interest rate:
[tex]\[ \text{Monthly interest rate} = \frac{r}{n} = \frac{0.04}{12} = 0.003333\overline{3} \][/tex]
2. Calculate the total number of deposits (compounding periods):
[tex]\[ \text{Total number of periods} = n \times t = 12 \times 30 = 360 \][/tex]
3. Apply these values to the future value formula:
[tex]\[ FV = 400 \times \left( \frac{(1 + 0.003333\overline{3})^{360} - 1}{0.003333\overline{3}} \right) \][/tex]
4. Compute the accumulated value:
- First, calculate [tex]\((1 + 0.003333\overline{3})^{360}\)[/tex]:
[tex]\[ (1 + 0.003333\overline{3})^{360} \approx 3.2434 \][/tex]
- Subtract 1 from this result:
[tex]\[ 3.2434 - 1 = 2.2434 \][/tex]
- Divide by the monthly interest rate:
[tex]\[ \frac{2.2434}{0.003333\overline{3}} \approx 673.02 \][/tex]
5. Finally, multiply by the deposit amount:
[tex]\[ FV = 400 \times 673.02 \approx 269,208 \][/tex]
Therefore, after 30 years, you will have approximately $269,208 in the account.
