Number Sequence Patterns
Sequences of positive integers often display patterns. For example:
\(\begin{align}1, 3, 5, 7, … \end{align}\)
is a pattern of odd numbers. It starts with 1, and each number is 2 greater than the previous number. Once you identify the pattern, you can predict the next number in the sequence. So in this example the next number is 9. This is an example of an Arithmetic Sequence.
Arithmetic Sequence
An arithmetic sequence, like the example above, starts with some number and then each succeeding number is a fixed amount greater or less than the previous number. Here are some examples:
\(\begin{align}2, 5, 8, 11, … \end{align}\)
\(\begin{align}12, 32, 52, 72, … \end{align}\)
\(\begin{align}10, 8, 6, 4, … \end{align}\)
In the last example each number is 2 less than the previous number.
Geometric Sequence
In a geometric sequence each number is multiplied by a fixed number to get the next number. Example:
\(\begin{align}3, 6, 12, 24, … \end{align}\)
This sequence starts with 3 and then each number is multiplied by 2 to get the succeeding number. Other examples:
\(\begin{align}4, 12, 36, 108, … \end{align}\)
\(\begin{align}10, -20, 40, -80, … \end{align}\)
Alternating Sequence
Sometimes a sequence is formed by first adding or subtracting, and then multiplying successive terms. For example:
\(\begin{align}1, 3, 9, 11, 33, 35, … \end{align}\)
We start with 1, and then we add 2 to get the next number, and multiply by 3 to get the one after that. The next 2 terms in the sequence are 105 and 107. More examples:
\(\begin{align}11, 15, 30, 34, 68, … \end{align}\)
Add 4, multiply by 2.
\(\begin{align}2, 1, 4, 3, 12, 11 … \end{align}\)
Subtract 1, multiply by 4.
Changing Differences
Sometimes the difference between numbers in a sequence is changing by a regular amount. For example:
\(\begin{align}1, 3, 7, 13, 21, … \end{align}\)
In this sequence we add 2 to the first number to get 3. Then we add 4 to get 7, and then add 6 to get 13, and then add 8 to get 21. The number we add to get the next term in the sequence is always 2 more than the number we added to get the previous term. More examples:
\(\begin{align}2, 5, 11, 20, … \end{align}\)
Add 3, then 6, then 9, etc.
\(\begin{align}3, 4, 7, 12, 19, 28, … \end{align}\)
Add 1, then 3, then 5, then 7, then 9, etc.
Fibonacci Sequence
A famous sequence is the Fibonacci sequence – named after a mathematician who lived around 1200. This sequence starts with two 1’s, and then each succeeding number is the sum of the two before it:
\(\begin{align}1, 1, 2, 3, 5, 8, 13, … \end{align}\)
You can see that the next numbers in this sequence are 21, 34, and 55. This sequence is useful in estimating the growth of populations – originally populations of rabbits.
Prime Number Sequences
These are difficult to recognize, because they don’t fit any of the patterns described above. For example:
\(\begin{align}41, 43, 47, 53, 57, … \end{align}\)
Here’s a hint: If a sequence doesn’t fit a normal sequence pattern, then you should suspect it’s a sequence in which all the numbers are prime. But how do you tell if a number’s prime? There is no easy fool-proof method, but if the number is less than 200, it will be prime if it’s odd and not divisible by 3, 5, 7, 11, or 13. Usually, this is good enough to recognize a prime number sequence. Here are some more examples:
\(\begin{align}53, 59, 61, 67, 71, 73, 79, …\end{align}\)
\(\begin{align}17, 19, 23, 29, 31, 37, 41, …\end{align}\)