$A = $200 \times \frac{\left[\left(1+\frac{0.055}{12}\right)^{(12 \cdot 20)}-1\right]}{\left(\frac{0.055}{12}\right)}$
Answer (2)
Calculate the term inside the parenthesis: 1 + 12 0.055 ≈ 1.0045833 .
Raise the result to the power of 12 × 20 = 240 : ( 1.0045833 ) 240 ≈ 2.99662556 .
Subtract 1: 2.99662556 − 1 = 1.99662556 .
Multiply by 200 and divide by 12 0.055 : A = 200 × 12 0.055 1.99662556 ≈ 87135.40 . The final answer is 87135.40 .
Explanation
Understanding the Formula We are given the formula for the future value of an annuity: A = 200 × ( 12 0.055 ) [ ( 1 + 12 0.055 ) ( 12 ⋅ 20 ) − 1 ] Our goal is to calculate the value of A . This formula calculates the future value of a series of regular payments.
Calculating the Inner Parentheses First, we need to calculate the value inside the innermost parentheses: 12 0.055 . The result of this division is approximately 0.0045833 . So we have: 1 + 12 0.055 ≈ 1 + 0.0045833 = 1.0045833
Calculating the Exponent Next, we calculate the exponent: 12 ⋅ 20 = 240 . Now we raise the result from the previous step to this power: ( 1.0045833 ) 240 ≈ 2.99662556
Subtracting 1 Now, we subtract 1 from the result: 2.99662556 − 1 = 1.99662556
Dividing by the Interest Rate Next, we divide the result by 12 0.055 ≈ 0.0045833 :
0.0045833 1.99662556 ≈ 435.677
Multiplying by 200 Finally, we multiply the result by 200: A = 200 × 435.677 ≈ 87135.40 Therefore, the value of A is approximately $87135.40 .
Examples
This formula is used to calculate the future value of an investment account where regular deposits are made. For example, if you deposit $200 per month into an account that earns 5.5% annual interest, compounded monthly, over 20 years, this formula tells you how much money you will have at the end of the 20 years. Understanding this calculation is crucial for financial planning, retirement savings, and investment strategies, helping individuals make informed decisions about their financial future.
The formula calculates the future value of an annuity based on regular monthly payments and compound interest. By following the calculation steps, the final amount accumulated after 20 years of monthly contributions is approximately $87134.80. This is essential for financial planning and understanding investment growth over time.
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