The Solution of a Difference Equation for Simple Interest Account

Basic Simple Interest Formula Recall that the new balance of an account that earns simple interest can be found by using the formula Bnew = Bprevious + rP . Recall the following: Bnew represents the new balance (which can also be thought of as the next balance) Bprevious represents the previous balance r represents the annual interest rate P represents the principal deposited (i.e. the initial deposit) into the account

Formula for a Future Balance of a Simple Interest Account A future balance of a simple interest account can be found by using the formula F = (1 + nr)P . Recall the following: F represents the future balance n represents the number of years r represents the annual interest rate P represents the principal deposited into the account The above formula allows us to find a future balance without having to find any of the previous balances.

The formula, F = (1 + nr)P , is a solution of the difference equation Bnew = Bprevious + rP. The following slides are a proof of the above statement.

Recall the General Form of a Difference Equation In chapter 11, a generalized difference equation (with an initial value) is written as: yn = a∙yn-1 + b , y0 where yn represents the next value in the list yn-1 represents the previous value in the list a is some value that is the coefficient of yn-1 b is some constant value y0 is the initial value

The formula Bnew = Bprevious + rP is a difference equation since it is in the form of yn = a∙yn-1 + b . The next slide will show this, but first we will rewrite Bnew = Bprevious + rP as Bnew = 1∙Bprevious + rP .

Bnew = 1 ∙ Bprevious + rP yn = a ∙ yn-1 + b where Bnew is equivalent to the yn Bprevious is equivalent to the yn-1 a = 1 b = rP

The Initial Value The initial value, y0 , for an account that earns simple interest is the principal, P (which is the initial deposit). Thus y0 = P .

Recall When Solving a Difference Equation Since a = 1 (see previous slide), then the solution of the difference equation can be found using yn= y0+nb . Now substituting the chapter 10 notations into the above formula

The Substitutions Substituting P for y0 and rP for b, the solution of yn= y0+nb becomes yn= P+nrP . Note that we did not change the notation of yn ; we will address this later.

Simplification Using Algebra The formula yn= P+nrP can be simplified to yn= (1+nr)P. We factored a P from both terms that are in the expression on the right side of the equal sign.

The yn Notation Change Now since yn , in yn= (1+nr)P , represents a specified value in the list (mainly the nth value in the list after the initial value) without using any of the previous values (with the exception of the initial value P), the yn can be changed to an F to represent the future value. So yn= (1+nr)P is changed to F = (1+nr)P.

is the solution of the difference equation Bnew = Bprevious + rP . Therefore F = (1+nr)P is the solution of the difference equation Bnew = Bprevious + rP . Q.E.D.