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Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this video lesson, we will learn about the many patterns that are possible in sequences. See how some sequences stop after a while and how some sequences never stop.

Sequences

What is a sequence? A sequence is a string of things in order. Because sequences have order, we can usually find a rule or pattern that fits. There is one famous sequence that we use almost every day of our lives. It is also one of the first things we learn. Can you think of what that is? Since we are talking math, think of numbers that are in order. Yes, it is our counting numbers. We begin by learning 1, 2, 3, and so on. We might learn to count to 10 first. But then we learn that these numbers actually continue forever. You could spend your entire life counting and never reach the end of the counting numbers.

Now let's talk about what kinds of sequences we can expect to see.

Finite Sequences

First, we have finite sequences, sequences that end. These sequences have a limited number of items in them. For example, our sequence of counting numbers up to 10 is a finite sequence because it ends at 10. We write our sequence with curly brackets and commas between the numbers like this: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. You can see that this sequence has order to it. It begins with a 1 and goes up by ones. Each number is the previous number plus 1.

We can also have a finite sequence made up of other things besides numbers. It can be an ordered sequence of letters, for example. Our alphabet is a finite sequence beginning with a and ending with z: {a, b, c, . . .x, y, z}. It has an order that we understand. We can also make a finite sequence out of names. For example, the name Sarah can be made into the sequence {s, a, r, a, h}, which contains all the letters in the name in order.

We can also create a sequence out of special groups of numbers. For example, we can have a finite sequence of the first four even numbers: {2, 4, 6, 8}. We can have a finite sequence such as {10, 8, 6, 4, 2, 0}, which is counting down by twos starting at 10.

The possibilities are endless. We can keep going and going.

Infinite Sequences

And this takes us to infinite sequences, which are sequences that keep on going and going. They have no end. For example, our counting numbers is an infinite sequence because it has no end. We can keep on counting forever. We write infinite sequences with the first few numbers to show the pattern and then with three periods in a row to show that it keeps going, like this: {1, 2, 3, 4...}.

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The possibilities here are endless as well. We can have an infinite sequence of alternating zeroes and ones: {0, 1, 0, 1, 0, 1...}. We can have an infinite sequence where each number is half of the previous number: {16, 8, 4, 2...}. We can have an infinite sequence of all the multiples of 5: {5, 10, 15, 20, 25...}.

Why It's Important

Why is it important to learn about sequences such as these? It is important to learn about these sequences, both the ones that end and the ones that don't, because understanding them will help us solve problems that involve these sequences. Math problems that involve addition, subtraction, multiplication, and division, for example, all require us to understand the sequence that is the number line. We need to know the order on the number line so we can add and subtract. Sometimes our problems will even ask us to write our sequences.

Sample Problem

For example, say you wanted to show your friend that growing his own orange tree would give him a lot of oranges. You could write a sequence. If the tree started out with zero oranges but the tree would produce 5 new oranges each day, your sequence would be {0, 5, 10, 15... 145} to show how many oranges would be on the tree each day. If you kept adding, you can show that by the end of a month - by day 30 - this orange tree would produce a lot of oranges!

Lesson Summary

Let's review what we've learned now. A sequence is a string of things in order. Finite sequences are sequences that end. Infinite sequences are sequences that keep on going and going.

Examples of finite sequences include the following:

The numbers 1 to 10: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Our alphabet: {a, b, c, . . . x, y, z}

The first four even numbers: {2, 4, 6, 8}

Examples of infinite sequences include these:

Our counting numbers: {1, 2, 3, 4...}

Alternating zeroes and ones: {0, 1, 0, 1, 0, 1...}

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