Fixed Annual Interest

Let's begin by coming up with a simple mathematical model of annual interest. Suppose that you invest P dollars in a bank account that pays annual interest at a rate of R. This means that at the beginning of the first year you have P dollars. After one year, the bank gives you an additional PR dollars, giving you a new balance of P + PR dollars. One year later your balance again increases by a factor of R, giving you

dollars. A year after this, you have

dollars. Can you generalize this process to determine how much money you'll have after n years?

Spotting the pattern in these increasingly longer formulas isn't necessarily easy. Let's see if Maple can help. Maple has a built-in function called factor that will factor a complex expression. Let's factor the expressions that give the amount of money after:

• one year

  factor(P + P*R);

• two years

  factor(P + 2*P*R + P*R^2);

• three years

  factor(P + 3*P*R + 3*P*R^2 + P*R^3);

A pattern that suggests a formula for the amount of money in the account after n years begins to emerge. What is it? Can you explain why it makes sense?

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