## Number Sequences

The **number sequence** is a set of numbers that show a series of a pattern.The term is the number in the sequence. There is a certain rule that a number follows, for example, 4, 8, 12 and this sequence shows that number 4 is added in each term. This is an example of an arithmetic sequence. We can easily identify the type of sequence used in a given by the operation such as addition, subtraction, division and multiplication.

*Example:
*2, 4, 6, 8, 10, …

1, 4, 9, 16, 25, …

1, 4, 8, 16, 32, ….

### Arithmetic Sequence

The rule of this sequence is to add or subtract the same value number each time. Using 4, 8, 12, we can easily see that the next term will be 16. *n* + 4 is the formula we will use to get the other sequence.

*Example:
*n + 4

4 The first term is 4, add 4 to get the second term.

4 + 4 = 8

8 + 4 = 12

12 + 4 = 16

Notice that for each term, 4 is being added. Another way to find the pattern is to compare the given and subtract each term. The given are 4, 8 and 12. Let’s compare each term:

12 – 8 = 4

8 – 4 = 4

Using this method, we know that 4 is the common difference in this term. Therefore, to get the fifth term we must add 4.

The __Linear sequence__ is often called an arithmetic sequence. The value of each term is increasing or decreasing by a definite amount.

*Example:
*9, 4, -1, -6, -11, …

9 – 4 = 5

4 – (-1) = 5

-1 – (-6) = 5

-6 – (-11) = 5

In this example, each of the terms decrease by 5. This is linear because it is decreasing in fixed amounts, which is 5.

### Geometric Sequence

This sequence rule is to divide or multiply the same number in each term.

*Example:
*3, 9, 27, ….

Find the pattern on each term. Observing each of the numbers, we can formulate a pattern. In observing, we can analyse the given starting with the last term and going into the first term. 3, 9, 27 have a common factor, which is 3. Using 3 as a multiple, we can form a formula in this pattern “3*n*”. Let’s work to get the next sequence. Let’s solve given the equation 3*n*.

3n

3 To get the second term using 3*n* = 3(3)

3 (3) = 9

3 (9) = 27

3 (27) = 81

Therefore the next in the sequence is 81. We cannot use an arithmetic sequence to get this equation if we subtract the last term to the term before, example, 27 – 9 = 18 and 9 – 3 = 6. There is no relation between the two answers.

*Example:
*Given

*n*(n + 2), if the first term is 1. Give at least 5 terms.

1st term: 1

2nd term: 1(1 + 2) = 3

3rd term: 3(3 + 2) = 15

4th term: 15(15 + 2) = 255

5th term: 255(255 + 2) = 65,535

The 5 terms of this sequence are 1, 3, 15, 255 and 65,535.

### Fibonacci Sequence

Fibonacci sequence is the series of numbers whose rules are that the next term is equal to the sum of the two numbers before. The Fibonacci sequence simply adds the first term and second term to get the third term, then adding the second and third terms to get the fourth term and so on. This is the Fibonacci sequence, 0, 1, 1, 2, 3, 5, 8, 13, … According to the rule, we can have this equation \(x_n=x_{n-1}+x_{n-2}\)

1 + 1 = 2

1 + 2 = 3

2 + 3 = 5

5 + 8 = 13

If we expand this series 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, … What will be the next term?

**Interesting Facts in Fibonacci Sequence
**Fibonacci sequences are often seen in nature. If we cut across an apple, it has 5 distinct sections and 3 sections in a banana. It can be seen in plants in petals: There are 3, 5, 13, or 21 petals. Sunflowers and even pine cones have Fibonacci numbers and in larger number in the Fibonacci sequence. If the last term divides to the term before, we will have the quotient of 1.618. 1.618 is called the ‘golden ratio’. In ancient Greece, the golden ratio is the number for physical perfection. In sculpture, painting and other works of art, it can be seen.

*Example:*

2583/1597 =1.618..

1597/987 = 1.618..