To calculate the future value of a series of annual deposits (also known as an annuity), we can use the future value of an annuity formula:
\[ FV = P \times \frac{(1 + r)^n - 1}{r} \]
Where:
- \( FV \) = future value of the annuity
- \( P \) = payment amount per period ($1000)
- \( r \) = annual interest rate (10% or 0.10)
- \( n \) = total number of payments (30 years)
Plugging in the values:
\[ FV = 1000 \times \frac{(1 + 0.10)^{30} - 1}{0.10} \]
Calculating step-by-step:
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Calculate \( (1 + r)^n \): \[ (1 + 0.10)^{30} = 1.10^{30} \approx 17.4494 \]
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Now substitute back into the formula: \[ FV = 1000 \times \frac{17.4494 - 1}{0.10} \] \[ FV = 1000 \times \frac{16.4494}{0.10} \] \[ FV = 1000 \times 164.494 = 164,494 \]
The future value of the Roth IRA after 30 years is approximately $164,494.
To find the total earned interest, we need to figure out how much was contributed over the 30 years:
- Total contributions = $1000 per year × 30 years = $30,000
Now subtract the total contributions from the future value to find the earned interest:
\[ \text{Total Earned Interest} = FV - \text{Total Contributions} \] \[ \text{Total Earned Interest} = 164,494 - 30,000 = 134,494 \]
Thus, the total earned interest is approximately $134,494.
In summary:
- Amount in the Roth IRA after 30 years: $164,494
- Total earned interest: $134,494
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