Alright, let's work through this step-by-step.
### Given:
- Monthly deposit (P): [tex]$400
- Annual interest rate (r): 5% or 0.05
- Compounding frequency (n): Monthly, which is 12 times a year
- Number of years (t): 35
### Let's break this down into the steps required to find the future value of this annuity:
1. Convert the annual interest rate to the monthly interest rate:
\[ \text{Monthly Interest Rate} = \frac{\text{Annual Interest Rate}}{\text{Compounding Frequency}} \]
\[ \text{Monthly Interest Rate} = \frac{0.05}{12} \]
\[ \text{Monthly Interest Rate} = 0.004167 \]
2. Determine the total number of compounding periods:
\[ \text{Total Periods} = \text{Compounding Frequency} \times \text{Number of Years} \]
\[ \text{Total Periods} = 12 \times 35 \]
\[ \text{Total Periods} = 420 \]
3. Use the Future Value of an Annuity formula:
The formula for the future value of an annuity is:
\[ \text{FV} = P \times \left( \frac{(1 + r)^{nt} - 1}{r} \right) \]
Where:
- \( P \) = Monthly deposit
- \( r \) = Monthly interest rate
- \( n \) = Compounding frequency
- \( t \) = Number of years
Substitute the given values into the formula:
\[ \text{FV} = 400 \times \left( \frac{(1 + 0.004167)^{420} - 1}{0.004167} \right) \]
4. Calculate the expression inside the parentheses:
\[ (1 + 0.004167)^{420} \]
5. Subtract 1 from the result:
\[ (1 + 0.004167)^{420} - 1 \]
6. Divide by the monthly interest rate:
\[ \frac{(1 + 0.004167)^{420} - 1}{0.004167} \]
7. Multiply by the monthly deposit:
\[ \text{FV} = 400 \times \left( \frac{(1 + 0.004167)^{420} - 1}{0.004167} \right) \]
After performing these calculations, we find that the future value (FV) is:
\[ \text{FV} \approx 454436.97 \]
### Answer:
In 35 years, you will have approximately $[/tex]454,436.97 in the account.
