Episode 14: April 15, 2010
by Jason Marshall
In the last few articles, we’ve talked about fractions and percentages, and soon enough we’ll see that these ideas naturally lead us into the world of decimal numbers. But before we head down that path, let’s take a quick detour to talk about what I consider to be a rather beautiful area of math—sequences and series. Today, we’ll discuss a particular type of sequence known as an arithmetic sequence. Then, in the weeks to come, we’ll take a look at geometric sequences, the famous Fibonacci sequence, and some truly fascinating mathematical series.
What is a Mathematical Sequence?
In both math and English, a “sequence” refers to a group of things arranged in some particular order. Outside of math, the things being arranged could be anything—perhaps the sequence of steps in baking a pie. But in math, the things being arranged are usually—no surprise here—numbers.
One example of a sequence is the list of numbers:
1, 2, 3.
Or, as an example of an entirely different sequence:
3, 2, 1.
Yes, both of these sequences have the same elements or members (1, 2, and 3), but they’re arranged in a different order—so they are, in fact, entirely different three-element long sequences. Of course, sequences don’t always have to have three elements—they can have any number of elements. For example:
2, 3, 5, 7, 11
is the sequence containing the first five prime numbers (those are natural numbers only divisible by themselves and 1). But why stop at five?—sequences can even be infinite! But how do you write something that’s infinitely long?
How to Write Mathematical Sequences
Okay, let’s briefly talk about the notation used to write sequences—including those that are infinitely long. First, the elements of a sequence are usually written out in a row, with each element separated by a comma. Sometimes the elements are grouped together inside parenthesis like
( 2, 3, 5, 7, 11 ),
but not always.
How to Write Mathematical Sequences That Are Infinitely Long
If a sequence has infinitely many elements, we indicate that by writing ellipses at the end of the sequence if it extends out indefinitely in the positive direction, or at the beginning of the sequence if it extends out indefinitely in the negative direction. For example, the sequence of positive integers can be written
1, 2, 3, 4, 5, …
The “…” indicates the sequence continues forever in the positive direction. The sequence of negative integers can be written
…, -5, -4, -3, -2, -1.
Here, the “…” indicates the sequence continues forever in the negative direction. Putting these two together, the sequence of all integers can therefore be written
…, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...
What are Arithmetic Sequences?
Now let’s talk about a specific type of mathematical sequence: the arithmetic sequence. I know it sounds complicated, but it’s really pretty simple. An arithmetic sequence is a sequence of numbers where the difference between any two successive elements is always the same constant value. For example, the sequence of years since the start of the new millennium is an arithmetic sequence:
2001, 2002, …, 2009, 2010.
Why is this an arithmetic sequence? Because the difference between all successive elements is always the same—2002 – 2001 = 1, 2010 – 2009 = 1—the difference is always 1.
There are two famous arithmetic sequences you’re already familiar with: the even and odd positive integers.
Notice I’ve used ellipses here in the middle of the sequence. What does that mean? Well, ellipses are used like this to represent missing elements—in this case: 2003, 2004, and so on, up to 2008. I could have written them all out explicitly, but using ellipses saves some writing.
What are Even and Odd Numbers?
The difference between successive elements in an arithmetic sequence doesn’t have to be 1—in fact, it can be anything. There are two famous arithmetic sequences you’re already familiar with whose successive members have differences of 2: the even and odd positive integers. Positive even integers begin at 2 and increase in steps of 2:
2, 4, 6, 8, 10, …
whereas positive odd integers begin at 1 and increase in steps of 2
1, 3, 5, 7, 9, …
Properties of Even and Odd Numbers
The members of these two sequences have some interesting properties. Whenever you add two even integers together, or two odd integers together, the answer is always an even number. For example, 2 + 6 = 8, 1 + 5 = 6, or 11 + 17 = 28—always even! However, whenever you add one even and one odd integer together, the answer is always odd. For example: 8 + 3 = 11 or 22 + 9 = 31—always odd!
Here’s a quick and dirty tip based upon this that can help you check your work: When you’re adding up numbers, you can use what’s called the “parity” of the numbers (that is, whether the numbers—or terms—you’re adding are even or odd), to make sure you have the right answer! If there are an even number of odd terms in your addition problem, the final answer must be even. However, if there are an odd number of odd terms in your problem, the final answer must be odd. For example, say you’re adding 23 + 6 + 79. Before even starting to add the numbers, I already know the answer must be even because there are an even number of odd terms (two, in this case: 23 and 79). This trick can be handy in everyday life, but it really shines when used on tests like the SAT or GRE to easily eliminate some of those multiple choices!
Next time, we’ll continue our tour of mathematical sequences with a look at geometric sequences. Until then, here’s a problem dealing with arithmetic sequences for you to contemplate:
Can you think of a more efficient way to fully define an arithmetic sequence other than simply writing out all its elements?
This one is a bit tricky. So think about it, and then look for the answer in this week’s Math Dude Video Extra! episode on YouTube and Facebook.
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Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!