# Recognizing & Generalizing Patterns in Math

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• 0:01 What is a Sequence?
• 1:05 Arithmetic & Geometric…
• 2:30 Sequences Based on Objects
• 4:08 Fibonacci Sequence
• 5:10 Lesson Summary
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Lesson Transcript
Instructor: Kevin Newton

Kevin has edited encyclopedias, taught middle and high school history, and has a master's degree in Islamic law.

Being able to recognize patterns is one of the most important aspects of studying math, and one that has definite advantages in other fields of study. In this lesson, we'll look at how to recognize the most common patterns in math: sequences.

## What Is a Sequence?

If I asked you to count by twos, what would you do? Would you give the screen a confused look, wondering if I was speaking a different language? Or would you intrinsically start to think '2, 4, 6, 8' in your head? Chances are you'd do the latter because you know the sequence of counting by twos. A sequence is a set of numbers in which the next number can be predicted from previous numbers. In other words, sequences are groups of numbers that follow a set progression. For people interested in math, sequences are one of the most interesting things around. There's a great deal of knowledge contained in them, and we're really only now starting to crack the code of some of the most advanced sequences. Still, we can use many of the more basic ones just about every day, much like counting in twos. In this lesson, we are going to look at two types of basic sequences, two more advanced sequences and then finally the most well-known of the truly advanced sequences. Don't worry, we'll do it in sequential order.

## Arithmetic & Geometric Sequences

When I asked you to count by twos, I was asking you to perform an arithmetic sequence. An arithmetic sequence is simply a sequence that is expanded or contracted by the same number of integers for each new term. Simple, right? In other words, you just add or subtract the same number over and over again. This number is called the common difference and is the key to establishing whether a sequence is arithmetic or not. It doesn't have to be a two, by the way, nor does it have to start at zero. As long as a sequence has the same common difference, it's arithmetic. As you can imagine, this is useful for everything from splitting a larger group into subgroups to figuring out percentages. For example, 2, 4, 6, 8 and so on is an arithmetic sequence.

Slightly more difficult are geometric sequences. Now, let's say that I just asked you to multiply a number by two over and over again. That would be a geometric sequence. That is when you simply multiply or divide the last number in a sequence to get the next number. The term for the quantity that you multiply the terms by is the common ratio. This is useful for predicting future events, like how far a stone will skip if tossed into water, and it reduces its distance by 50% with each skip it makes. For example, 2, 4, 8, 16, 32 and so on is a geometric sequence.

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