## Sequences of Numbers

One of the important types of patterns in mathematics is sequences of numbers--lists of numbers where you can tell what the next number is if you know the rule.

Let me give you a few examples.

**Examples of Sequences**
Rule | Sequence |

Add 1 to the previous number | 1, 2, 3, 4, 5, . . . |

Write only the odd numbers | 1, 3, 5, 7, 9, . . . |

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

Add 3, then subtract 1, then repeat | 1, 4, 3, 6, 5, . . . |

Being able to discover patterns is an important part of mathematics and a useful skill in almost any job.

### Puzzle 1

Find the next two numbers in each of the sequences in the table above.

### Puzzle 2

Here are some sequences for you to investigate. See if you can find a rule for each and determine what the next two numbers are.
A. (This is about as easy as they come): 1, 1, 1, 1, 1, . . .

B. (This is similar to one of the examples): 1, 4, 7, 10, 13, . . .

C. (For this one, look at the differences between the numbers in the sequence): 1, 2, 4, 7, 11, 16, . . .

D. (This is a famous sequence, called the Fibonacci numbers. To find the rule for it compare each number with the two numbers preceding it.):1, 1, 2, 3, 5, 8, 13, . . .

E. (To find the rule for this one, compare each number to the positive integers (counting numbers)):1, 4, 9, 16, 25, 36, . . .

### Puzzle 3

Here are a couple of sequences that don't use arithmetic.
F. (It may be useful to write out the numbers one, two, three, and so on.): 3, 3, 5, 4, 4, 3, . . .

G. (Try Roman numerals -- e.g. I, II, III, IV, V etc. --to get started on this one): 1, 2, 3, 3, 2, 3, . . .

H. (This one is called the E-ban numbers. The name is a hint. Again write out the numbers one, two, etc. and compare those included with those excluded.) 2, 4, 6, 30, 32, 34, 36, 40, . . .

Finally some very important sequences:

### Puzzle 4

I. This sequence is very important in mathematics: 2, 3, 5, 7, 11, 13, 17, . . .
J. Sequences can include fractions: 2, 1, 1/2, 1/4, 1/8, 1/16, . . .

K. and decimals (You may have to look this up to find the next numbers in the sequence): 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592, 3.1415926, . . .