# At a Glance - Arithmetic, Geometric & Exponential Patterns

You have actually been working with algebra since you were three and began to notice patterns (red dog, blue cat, red dog, blue cat…). The patterns we are going to work with now are just a little more complex and may take more brain power. Patterns are the beginning of algebra.

There are endless types of patterns and methods for solving patterns. You mostly have been working with simple arithmetic (patterns involving adding or subtracting a number each time) or geometric patterns (ones involving multiplying or dividing by a number).

Here are three common types of patterns you may have seen.

Type | Example | Solution |
---|---|---|

Arithmetic | 1,3,5,7,9... | Add 2 each time. |

99,90,81,72... | Subtract 9 each time | |

Geometric | 1,2,4,8,16... | Multiply the previous number by 2. |

1000,100,10,1... | Divide the previous number by 10. | |

Geometric - Exponential | 2,4,16,256... | Square the previous number. |

1,4,9,16,25... | 1^{2},2^{2},3^{2},4^{2},5^{2}... |

**Look Out**: an exponential pattern is actually a type of geometric pattern. However, to help explain things, we made them a subcategory.

Here's a video example of math in action. Just a sample of questions you might see.

#### Example 1

The first four triangle numbers are 1, 3, 6, and 10. They are called triangular because they can be arranged in dots as triangle, like so: What will the 10th triangular number be? |

#### Example 2

Find the next two numbers in this pattern: 1, 2, 8, 48, 348… Some of you may have figured out the pattern by just looking at it, the rest of us may need a little help. Let's start by writing these numbers in a chart and looking at their differences. |

#### Example 3

Find the missing number in the pattern: 3, 9, 81, ___, 43046721. As mentioned before, there are three basic types of patterns: arithmetic, geometric, and geometric exponential. Let's try all three and see if one fits. Here you can see that geometric-exponential patterns are also geometric. We can use either of these patterns to fill in the blanks. |

#### Exercise 1

Find a pattern then fill in the next two numbers: 0, 1, 5, 14, 30, ____, ____

#### Exercise 2

A triangle has no diagonals, while a quadrilateral has 2, a pentagon has 5, and a hexagon has 9. Without drawing the figure, how many diagonals will a septagon (7-sided figure) have?

#### Exercise 3

The numbers 1–9 are arranged in a certain order. Find the missing numbers.8 5 4 ___ 1 7 6 ___ ___