4.2

# Examples of sequences

Below are some examples of number sequences, some have a mathematical rule behind them, others do not.

## Challenge

Take a look at the following sequences.

a) 3, 3, 5, 4, 4, 3, 5, 5, _, _, …
b) 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, …
c) 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, _, _, …
d) 3, 7, 11, 15, …
e) 2, 4, 8 ,16, 32, …

Can you spot any patterns in them? Can you find a rule that generates them?

## Solutions

There are many kinds of number sequences. The first two in our list are not mathematical sequences. Whilst we can describe how to find each number in the sequence we cannot find a mathematical rule.

### a) 3,3,5,4,4,3,5,5,,

The next two numbers in this sequence are 4 and 3. Can you see why? The sequence is the number of letters in the words ONE, TWO, THREE, FOUR, FIVE and so on… it is a definitely a fun sequence but not very mathematical.

### b) 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, …

Did you say this one out loud? This is the ‘look and say’ sequence. Each term describes the term before it:

• 1 is read off as “one 1”
• 11 is read off as “two 1s”
• 21 is read off as “one 2, then one 1” and so on

There are many number sequences we can make up, often used in quizzes or shared on social media, which are often interesting but may not have much mathematical value.

The sequences we will concentrate upon in this course are some of the mathematical sequences found in the school curriculum.

### c) 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, _, _,

Is the Fibonacci sequence. We will explore this sequence in more detail later in the week.

### d) 3, 7, 11, 15, …

Is an example of an arithmetic sequence where the difference between each numbers the same.

### e) 2, 4, 8 ,16, 32, …

Is an example of a geometric sequence where we multiply by the same number each time to get the next number.

## Problem worksheet

Now complete question 1 from this week’s worksheet.