Both a Borrower and a Lender Be

Both a Borrower and a Lender Be

Author(s): William Miller

The College Mathematics Journal, Vol. 16, No. 4 (Sep., 1985), p. 284.

Published by: Mathematical Association of America

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Tags: First order equations

Abstract: This Paper provides a great example of how important it is to pay attention to your money, and your interest rates. This article states that it is well known that the initial amount, $P_0$, annual interest rate $r$, compounded $m$, and time, $t$, while earning annual interest is equal to $P(t) = P_0(1+r/m)^{mt}$. It is said that as time goes to infinity you will get $dP/dt = rP$. We learned however, that calculus has failed to inform us that this formula isn’t all true for a change in time. The first payment is to cover the cost from the accumulation of interest. After the first payment the balance is now $P_1 = P_0(1+r/m)-k/m$. This new information allows the person with the payment plan to know when to exceed the interest due, and what is the result of that. This lets us know that by summing the geometric series, and knowing how this series will react over time, we will know that as m, or times compounded per year, goes to infinity we will get, $P(t) = k/r - (k/r – P_0)e^{rt}$. You will notice that this is the solution to our original differential equation, $dP/dt = rP- k$. Using the differential equation, we obtained more information about this equation, like how it will react over time, or that added k term that wasn’t in the equation before. Steven F. Choate, Stephen F. Austin State University, May 3, 2017.

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