Final answer:
The future value of a $100 investment at a continuously compounded interest rate of 7.5% for 7 years can be calculated using the formula A = Pe^(rt). Substituting the given values into the formula, you can determine the amount to which the initial investment will grow.
Step-by-step explanation:
To calculate the future value of $100 when it is invested at a continuously compounded interest rate of 7.5% for 7 years, we can use the formula for continuous compounding: A = Pert, where A is the amount of money accumulated after n years, including interest, P is the principal amount (the initial amount of money), r is the annual interest rate (decimal), and t is the time in years. In this case, P = $100, r = 0.075, and t = 7 years.
Plugging these values into the formula yields:
A = 100e(0.075×7)
Once you calculate the expression using a calculator with an exponential function, you will find the total amount to which the $100 will grow after 7 years.
The power of compound interest is evident when comparing to simple interest, which only calculates interest on the principal. Over time, compound interest can significantly increase the growth of an investment, which makes starting to save money early in life a wise financial decision.