Arithmetic, Geometric, and Exponential Patterns Examples

The first four triangle numbers are 1, 3, 6, and 10. They are called triangular because they can be arranged in dots as triangles, like so:

What will the 10th triangular number be?

There are many ways to work out a pattern. We like to start by arranging the numbers in a row to see how they relate. Underneath each number write what needs to be added, multiplied, or divided to get the next number.

Notice a pattern? The number 5 would be added next, then 6, then 7 and so on.

So, the 10th triangular number is 55.

Find the next two numbers in this pattern: 1, 2, 8, 48, 384…

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.

Hmm… addition doesn't seem to be working, so let's try multiplication.

Seeing the pattern yet? Each time we multiply by the next even number. Here is how it would continue:

The next two numbers in the pattern are 3840 and 46,080.

Find the missing number in the pattern: 3, 9, 81, ___, 43,046,721.

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 we can see that geometric-exponential patterns are also geometric. We can use either of these patterns to fill in the blank, which gives us our missing number: 6561.