Final answer:
The balance of an account after 7 years with $200 invested at an APR of 14% with annual compounding is approximately $551.82. The balance of an account with $222 invested at an APR of 2% compounded annually after 538 years would be a mathematically interesting but impractical figure given the time span.
Step-by-step explanation:
The subject of this question relates to the calculation of future values of investments using the compound interest formula A = P(1 + r/n)^(nt), where:
- P is the principal amount (initial deposit)
- r is the annual interest rate (decimal)
- n is the number of times that interest is compounded per year
- t is the time the money is invested or borrowed for, in years
- A is the amount of money accumulated after n years, including interest.
For part a, to find the balance of an account after investing $200 at a 14% APR with annual compounding after 7 years, we use the formula:
A = 200(1 + 0.14/1)^(1*7)
A = 200(1 + 0.14)^7
A = 200(1.14)^7
A = 200(2.75911)
A = $551.822
After rounding to the nearest cent, the balance would be $551.82.
For part b, to find the balance of an account after investing $222 at a 2% APR with annual compounding after 538 years, we use the formula:
A = 222(1 + 0.02/1)^(1*538)
A = 222(1.02)^538
Here, since the time is impractically long and the amount would be incredibly high, we observe the power of compound interest over extended periods. However, for practical purposes, this kind of calculation is hypothetical given that no investment situation would likely last for 538 years.