$A=200 \times\left[\frac{\left(1+\frac{5.5}{12}\right)^{(12 \cdot 20)}-1}{\frac{5.5}{12}}\right]$
Answer (2)
Calculate the value inside the parenthesis: 1 + 12 5.5 = 1.458333... .
Calculate the exponent: 12 × 20 = 240 .
Raise the result from the first step to the power calculated in the second step: ( 1.458333... ) 240 = 2.116569985781438 × 1 0 39 .
Calculate the final value of A: A = 200 × 12 5.5 ( 1.458333... ) 240 − 1 = 9.235715936778864 × 1 0 41 .
The value of A is 9.235715936778864 × 1 0 41 .
Explanation
Understanding the Formula We are given the formula A = 200 × [ 12 5.5 ( 1 + 12 5.5 ) ( 12 × 20 ) − 1 ] . This formula calculates the future value of an annuity, where:
The principal payment is 200.
The interest rate is 5.5% per year.
The number of times interest is applied per time period is 12.
The number of time periods elapsed is 20.
Our objective is to calculate the value of A.
Calculate the Value Inside Parenthesis First, we need to calculate the value inside the parenthesis: 1 + 12 5.5 .
1 + 12 5.5 = 1 + 0.458333... = 1.458333...
Calculate the Exponent Next, we calculate the exponent: 12 × 20 .
12 × 20 = 240
Raise to the Power Now, we raise the result from Step 2 to the power calculated in Step 3: ( 1.458333... ) 240 = 2.116569985781438 × 1 0 39
Subtract 1 Subtract 1 from the result of Step 4: 2.116569985781438 × 1 0 39 − 1 = 2.116569985781438 × 1 0 39 (Since 1 is negligible compared to 2.116569985781438 × 1 0 39 )
Divide by Fraction Divide the result of Step 5 by 12 5.5 :
12 5.5 2.116569985781438 × 1 0 39 = 0.458333... 2.116569985781438 × 1 0 39 = 4.617857968389432 × 1 0 39
Multiply by 200 Finally, multiply the result of Step 6 by 200 to obtain the final value of A: A = 200 × ( 4.617857968389432 × 1 0 39 ) = 9.235715936778864 × 1 0 41
Final Answer Therefore, the value of A is: A = 9.235715936778864 × 1 0 41
Examples
Understanding annuities is crucial for financial planning. For instance, if you invest $200 monthly with a 5.5% annual interest rate compounded monthly for 20 years, this formula helps calculate the future value of your investment. This calculation is essential for retirement planning, understanding loan repayments, and evaluating investment options. By mastering this concept, you can make informed decisions about your financial future and ensure long-term financial security. It provides a foundation for understanding compound interest and its impact on savings and investments.
The formula calculates the future value of an annuity where $200 is deposited monthly at a 5.5% annual interest rate for 20 years. After performing the necessary calculations, the final value of A is approximately 9.2357 × 1 0 41 . This represents the compounded growth of the initial investment over the specified period based on the given rate and frequency of compounding.
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