Chapter 5, Lesson 2

In this lesson students will:

- recognize the form of a geometric series
- determine the sum of a geometric series using the closed-form formula
- use a time diagram to develop the future value formula
- calculate the future value of an ordinary simple annuity

## Future Value – Ordinary Simple Annuities

#### Geometric Series

In this lesson, our goal is to develop the future value formula for a series of equivalent payments that all earn compound interest over time. To fully understand how the formula is developed, we must first briefly look at geometric series.

If a sum of numbers has a certain format it is known as a *geometric series*. The format of the numbers being added together begins with the number and then all numbers following are multiplied by the same number, called the *common ratio* . Let’s refer to the sum of the geometric series as . The format can then be summarized as follows, where we take as the power of the last term for convenience:

Next, we would like to develop a formula for calculating the sum easily. To do this, we can multiply all the terms in our formula above by the common ratio and then subtract the original sum .

Finally, by factoring the left side of our equation and dividing by we obtain a closed-form formula for finding the sum of a geometric series.

The video below will guide you through two examples of how to use the closed-form formula.

#### The Time Diagram Approach

Next, let’s turn our attention to setting up a time diagram to develop the future value formula. We would like to consider an ordinary simple annuity. This means that equivalent payments, which we label as , occur at the end of each compound period; and that compound frequency is the same as the payment frequency. To denote the time periods in which compounding and payments occur, we use . The video below will guide you through the formulation of the future value formula for an ordinary simple annuity.

#### Future Value Formula Examples

As we saw in our previous video, the future value formula for an ordinary simple annuity is:

Let’s work through a couple examples to help solidify our use of the formula.