# Pattern Recognition

When solving math problems, it is often good if we are able to recognize patterns in the numbers that we are seeing. This will allow us to hypothesize what the general term of the sequence will look like, which could help to guide the approach for the problem.

You should be aware of sequences like the integers, odd numbers, perfect squares, primes, factorials, exponents, etc, and these should be easily identified.

Trying to determine the pattern from a sequence of numbers can be considered an art. There is no strict rules that we can follow which will guarantee a result. We can merely use our powers of observation and intuition to guide our guesses. Some common approaches are to look at successive differences between terms, adding another sequence (often constant or linear) to the current numbers, dividing out by any common factors, etc.

The On-Line Encyclopedia of Integer Sequences is a good reference that you could use, if you are confronted with a mathematical sequence and are unable to recognize the pattern. It is the largest database of integer sequences, storing over 220, 000 sequences that are of interest to mathematicians and amateurs. It was started by Neil Sloane as a graduate student in 1965, to help him recognize sequences that arose from his work in combinatorics.

## Examples

### 1. What comes next: $1, 3, 5, 7, \_$?

Solution: We recognize this as the list of odd numbers, and so the next term is 9. $_\square$

### 2. What comes next: $2, 3, 5, 7, \_$?

Solution: We recognize this as the list of primes, and so the next term is 11. $_\square$ Note by Arron Kau
6 years, 8 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$