Topic 7: Sequence and Number Patterns

Sequence and Number Patterns

Numbers can have interesting patterns. 

Here we list the most common patterns and how they are made.

Arithmetic Sequences

An Arithmetic Sequence is made by adding the same value each time.

Example:         

                                                1, 4, 7, 10, 13, 16, 19, 22, 25, ...

This sequence has a difference of 3 between each number.
The pattern is continued by adding 3 to the last number each time, like this:

Example:

3, 8, 13, 18, 23, 28, 33, 38, ...

This sequence has a difference of 5 between each number.
The pattern is continued by adding 5 to the last number each time, like this:

The value added each time is called the "common difference"

What is the common difference in this example?

19, 27, 35, 43, ...

Answer: The common difference is 8

The common difference could also be negative:

Example:

25, 23, 21, 19, 17, 15, ...

This common difference is −2
The pattern is continued by subtracting 2 each time, like this:

What we multiply by each time is called the "common ratio".

In the previous example the common ratio was 3:

We can start with any number:

Example: Common Ratio of 3, But Starting at 2

2, 6, 18, 54, 162, 486, ...

This sequence also has a common ratio of 3, but it starts with 2.

Example:

1, 2, 4, 8, 16, 32, 64, 128, 256, ...

This sequence starts at 1 and has a common ratio of 2.

The pattern is continued by multiplying by 2 each time, like this:

The common ratio can be less than 1:

Example:

10, 5, 2.5, 1.25, 0.625, 0.3125, ...

This sequence starts at 10 and has a common ratio of 0.5 (a half).

The pattern is continued by multiplying by 0.5 each time.

But the common ratio can't be 0, as we would get a sequence like 1, 0, 0, 0, ...

Special Sequences

Triangular Numbers

1, 3, 6, 10, 15, 21, 28, 36, 45, ...

This Triangular Number Sequence is generated from a pattern of dots which form a triangle.

By adding another row of dots and counting all the dots we can find the next number of the sequence:

Square Numbers

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, ...

They are the squares of whole numbers:

0 (=0×0)
1 (=1×1)
4 (=2×2)
9 (=3×3)
16 (=4×4)

Cube Numbers

1, 8, 27, 64, 125, 216, 343, 512, 729, ...

They are the cubes of the counting numbers (they start at 1):

1 (=1×1×1)
8 (=2×2×2)
27 (=3×3×3)
64 (=4×4×4)

QUESTIONS 

[1] What is the next number of the sequence 1, 3, 7, .... , given that the rule for the sequence is:
x₁ = 1
x_n = x_n-1 + 2^n-1 , n ≥ 2

[2] What is the next number of the sequence 0, 2, 5, 9,... , given that the rule for the sequence is:

x_n = ½(n² + n - 2)

[3] Use differences to find the rule for the sequence {0, 2, 6, 12, 20, ...}

[4] Use differences to find the rule for the sequence {1, 5, 14, 28, 47, ...}

[5] Use differences to find the rule for the sequence {3, 11, 24, 42, 65, ...}