Sequence & Number Pattern Part 3

Arithmetic Sequences

In an Arithmetic Sequences the difference between one term and the next is a constant.

In other words, 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. 

Its Rule is x_n = 3n-2

The pattern is continued by adding 3 to the last number each time, like this:

In General we can write an arithmetic sequence like this:

{a, a+d, a+2d, a+3d, ... }

where:

·         a is the first term, and

·         d is the difference between the terms (called the "common difference")

 And we can make the rule:

x_n = a + d(n-1)

(We use "n-1" because d is not used in the 1st term).

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"

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:

Geometric Sequences

In a Geometric Sequences each term is found by multiplying the previous term by a constant.

Example:

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

This sequence has a factor of 2 between each number.
Its Rule is x_n = 2ⁿ

Example:

1, 3, 9, 27, 81, 243, ...

This sequence has a factor of 3 between each number.
The pattern is continued by multiplying by 3 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, …

In General we can write a geometric sequence like this:

{a, ar, ar², ar³, ... }

where:

·         a is the first term, and

·         r is the factor between the terms (called the "common ratio")

Note: r should not be 0.

When r=0, we get the sequence {a,0,0,...} which is not geometric

And the rule is:

x_n = ar^(n-1)

(We use "n-1" because ar⁰ is the 1st term)

Reference:

http://www.mathsisfun.com/algebra/sequences-series.html