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When discussing continuous compounding, we venture into a scenario where interest is added so frequently that it is essentially being compounded constantly – at every moment in time. Mathematically, this constant growth is modeled using the natural exponential function, denoted as 'e'. This function represents unending compound interest.

The formula for continuous compounding is: \[ A = Pe^{rt} \] where 'A' is the future value of the investment/loan, including interest, 'P' is the principal investment amount, 'r' is the annual interest rate (decimal), 't' is the time the money is invested or borrowed for, and 'e' is approximately equal to 2.71828. In Option 2 from the exercise provided, the bank uses continuous compounding, which often results in slightly more interest earned over time compared to traditional compounding methods due to the frequency of compounding.

In contrast to continuous compounding, when interest is compounded quarterly, it means the interest is calculated and added to the principal four times a year – at the end of every quarter. The formula to calculate quarterly compounding is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] 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), 'n' is the number of times that interest is compounded per year, and 't' is the time the money is invested in years. For Option 1, the bank compounds interest four times a year (quarterly), which means 'n' would be 4 in our formula.

The interest calculation for both options involves determining the amount of money that the investment will yield over the specified period. After finding the future value 'A', we subtract the principal 'P' to find the total interest earned. This process involves careful attention to the specifics of the compounding method used and accurate substitution into the appropriate formula. For example, Option 1 with quarterly compounding requires us to divide the annual rate by the number of periods per year, raise this to the total number of compounding periods, and multiply by the principal. Meanwhile, continuous compounding in Option 2 utilizes the exponential 'e' to simulate that the investment grows infinitesimally in every instant.

For Samuel's scenario, correct interest calculation using the formulas mentioned above will help him accurately compare the interest earned from both options and make an informed decision.

At the intersection of finance and mathematics lies financial algebra, a field that involves a variety of functions and formulas to solve real-world monetary problems. The compound interest formulas used in the exercise for quarterly and continuous compounding are examples of financial algebra in action. The formulas describe how investments grow over time, a fundamental concept of savings, loans, and investments.

This discipline not only deals with compound interest, but also teaches about various financial instruments, annuities, saving plans, loan repayments, and more, giving a broad overview of personal and business finances. Understanding financial algebra enables students and professionals to make prudent financial decisions by analyzing how money will grow or shrink over time under various conditions and interest rates.