Sequences -- Quickstart

You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.

There are many situations that can be best described by a sequence of numbers -- for example, suppose that you invest $100.00 in a bank account that earns 4% interest each year. We usually denote the original investment -- the amount of money in the account at the beginning of the first year -- by p(1). For this example, p(1) is $100.

At the beginning of the second year, you will have earned $4.00 in interest bringing the total amount in the account to $104.00. We usually denote this p(2). Notice that

p(2) = p(1) + 0.04 * p(1) = 1.04 * p(1)

At the beginning of the third year the total amount in the account will be

p(3) = p(2) + 0.04 * p(2) = 1.04 * p(2) = $108.16.

We continue in the same way. The amount of money in the account at the beginning of each subsequent year is denoted p(4), p(5), p(6), and so forth.

The equation

p(n) = 1.04 * p(n - 1)

provides a general rule for computing the change in the amount of money in the account from one year to the next. You can think of this equation as describing how to compute the bank balance in year n from the bank balance the preceding year, year n - 1. The same rule for change can be written

p(n + 1) = 1.04 * p(n)

which can be thought of as describing how to compute next year's bank balance from this year's bank balance.

Use your CAS window to verify the calculations above. Then compare two different bank accounts -- one starting with $200.00 and earning 3% interest and the other starting with only $100 but earning 5% interest. How long would it take until the second bank account catches up with the first one? answer

These are typical examples of sequences of numbers, each number describing the amount of money in the account at a particular time and collectively describing the past and future history of the account. Sequences are used for many purposes -- for example, describing the past and future history of prices for a particular product or the past and future history of the population of a particular species in a particular habitat.

The notation p(1), p(2), p(3), ... is quite common. However, other notations are also used. For example, Mathematica uses the notation p[1], p[2], and human beings often use subscripts, writing, for example,

Missing Equation

Sometimes it is convenient to number the terms in a sequence differently. For example, if we are looking at the amount of money in a bank account we might begin the sequence using the notation p(0) instead of p(1) for the initial deposit, or we might use the actual year of the first deposit -- for example, p(1995) .

Particular sequences may be defined in many different ways -- for example,

By a physical measurement -- for example, p(1) might be the temperature measured in the shade at 12:00 noon on January 1, 1995 at the Salt Lake City airport; p(2) might be the temperature measured in the shade at 12:00 noon on January 2, 1995 at the Salt Lake City airport; and so forth.
By an economic or financial measurement -- for example p(1) might be the wholesale price of a AA alkaline battery on January 1, 1995; p(2) might be the price of the same battery on January 1, 1996; and so forth.
Geometrically -- for example p(n) might be the area of a square whose sides are n feet long.
By an algebraic expression -- for example
```
p(n) = n^2 feet
```
Notice that numerically the last two examples are the same. The formula for the area of a square whose sides are n feet long is
```
Area = n^2 feet
```
Even though these two descriptions describe the same sequence of numbers they tell us different things. The first description describes the sequence in terms of geometric concepts -- squares and area -- while the second description tells us how to compute the sequence arithmetically.
The amount of money in a bank account earning 4% interest is typically described by a pair of equations -- for example,
```
p(1) = 12,000
p(n + 1) = 1.04 p(n)
```
This kind of description is particularly useful. It describes two aspects of the sequence
- How it starts -- the equation
```
p(1) = 12,000
```
- and how it changes -- the equation
```
p(n + 1) = 1.04 p(n)
```
An equation like the first equation, describing how a sequence starts, is often called an initial condition. An equation like the second equation, decsribing how a sequence changes is often called a change equation. Notice that this rule for change could just as well have been writtten
```
  p(n) = 1.04 * p(n - 1)  
```

When the population in a particular habitat is growing because of immigration it might increase by a fixed amount each year. For example, we might have


p(1) = 10,000
p(n) = p(n - 1) + 500

This same model can be described by algebraic formula


p(n) = 10,000 + 500(n - 1)

Use your CAS window to check that these two descriptions produce the same sequence of numbers by computing p(10), p(17), and p(27) using the two different descriptions. This is not a proof that the two descriptions produce the same sequence of numbers but it does provide some evidence that they do. answer

Check Your Understanding

The volume of a cube whose sides are n feet long. answer
The amount of money in a bank account earning 3% interest, n years after an initial deposit of $1000.00. answer
Suppose the price of a particular textbook is $30.00 in 1995 and the price rises $3.00 each year. Let p(1) be its price in 1995; p(2) its price in 1996; and so forth. answer
Suppose the price of a particular textbook is $30.00 in 1995 and the price rises 10% each year. Let p(1) be its price in 1995; p(2) its price in 1996; and so forth. answer
Describe each of the following sequences using an initial condition and a change equation.
The amount of money in a bank account earning 3% interest, n years after an initial deposit of $1000.00. answer
Suppose the price of a particular textbook is $30.00 in 1995 and the price rises $3.00 each year. Let p(1) be its price in 1995; p(2) its price in 1996; and so forth. answer
Suppose the price of a particular textbook is $30.00 in 1995 and the price rises 10% each year. Let p(1) be its price in 1995; p(2) its price in 1996; and so forth. answer

Stretch Your Understanding

Copyright c 1995 by PWS Publishing Company, a division of International Thomson Publishing Inc. Comments to Frank Wattenberg, Department of Mathematics, Carroll College, Helena, MT 59625.