Final answer:
The question deals with compound interest, which is a critical concept in financial mathematics. It illustrates how periodic deposits grow over time due to quarterly compounding. Examples provided demonstrate compound interest's effects over various periods and initial investments.
Step-by-step explanation:
The question is focused on the subject of compound interest which is a key concept in Mathematics, particularly in financial math. It relates to the scenario in which Danielle makes periodic deposits into a savings account that grows due to the interest compounding quarterly.
Compound interest is the interest calculated on the initial principal and also on the accumulated interest of previous periods of a deposit or loan. Compound interest can be calculated using the formula A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate (decimal), n is the number of times that interest is compounded per year, and t is the time the money is invested for in years.
In the examples provided, we see the power of compound interest over time. For instance, with a $100 investment at 2% compound interest, you end up with $115.76 after three years, which is slightly more than with simple interest due to compounding. Meanwhile, a $3,000 investment at a 7% annual rate over 40 years grows nearly fifteen fold. These examples emphasize the significance of the power of compound interest and the benefits of starting to save early.