Overview:
One problem-solving strategy in mathematics is to look for a pattern that will predict the next number in a sequence. That is often the first step to writing an equation that describes how to find each number. Some famous patterns, such as Fibonacci numbers and Pascal’s triangle, play an important role in number theory.
What Are Arithmetic Patterns?
Arithmetic patterns are those patterns that can be predicted by adding a constant number to each number in the sequence. The sequence that begins {1, 2, 3, 4, 5…}, the natural numbers, is generated by adding 1 to each number. Similarly the pattern that begins {2, 4, 6, 8…} and the one that begins {1, 3, 5, 7…} are generated by adding 2 to each number. Those simple patterns follow an arithmetic sequence. Similarly, the pattern that begins {3, 7, 11, 15 …} is generated by adding 4 to each number.
What Are Geometric Patterns?
Geometric patterns can be predicted by multiplying each number in the sequence by a constant. For example, the sequence that begins {5, 10, 20, 40…} is generated by multiplying each number by 2. Similarly, the sequence that begins {4, 2, 1, 0.5, 0.25…} is generated by multiplying each number by ½ (or dividing by 2).
What Are Mixed Patterns?
Some number patterns can combine arithmetic and geometric sequences, or follow other rules entirely. Suppose the sequence is {1, 3, 6, 8, 16…}. The next number in the sequence is 18, and the one following that is 36. That is because 1 + 2 is 3, times 2 is 6, + 2 is 8, times 2 is 16, + 2 is 18, times 2 is 36. Similarly, a pattern such as {1, 3, 3, 5, 3, 7, 3…} is 1 + 2, repeat 3, 3 + 2, repeat 3, 5 + 2, repeat 3, and so on, so that the next numbers in the series would be 9, 3.
What Are Some Special Patterns?
The Fibonacci numbers are a special pattern. They are in the sequence {0, 1, 1, 2, 3, 5, 8, 13, 21…}, and have properties of their own. Prime numbers are in the pattern {1, 2, 3, 5, 7, 11, 13…}. The pattern {1, 4, 9, 16, 25, 36…} are the squares.
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