###### The Arithmetic Sequence

Maths Applications (Year 12) - Growth and Decay in Sequences

Bhakti Sharma

# The Arithmetic Sequence

Arithmetic Sequences are those in which each term is obtained by a constant number being added to the previous term. Arithmetic Sequences are also known as Arithmetic Progressions in some textbooks, or APs for short. An example would be 2, 4, 6, 8, 10..., in which the next term is always obtained by adding 2 to the previous term. In this example, the first term is 2 and the common difference is 2, as well.

Let's go through some examples:

1, 3, 5, 7, 9, ...

The first term is 1 and the common difference is 2

5, 10, 15, 20, 25, ...

The first term is 5 and the common difference is 5

30, 26, 22, 18, 14, ...

The first term is 30 and the common difference is -4

As the examples above show, every arithmetic sequence is in the form of:

a, a + d, a + 2d, a + 3d, a + 4d, a + 5d, ...

In which a is the first term, and d is the common difference

To define this recursively, we can say:

T n+1 = Tn + d, T1 = a

Some textbooks might also use a lowercase T instead.

Let's go through an example:

Find the first four terms of a sequences if T n+3 = Tn + d, T1 = 2

The general form tells as that the common difference is 3 and that the first term is 2, so:

The first four terms are 2, 5, 8 and 11.

Let's say that a questions asks you to create a general form, given that you have a sequence of numbers.

30, 28, 26, 24, 22, 20,..

Here we can see that the first term is 30, so T1 = 30. We can also see that there is a difference of -2 between each consecutive term of the sequence, hence:

T n+1 = Tn - 2, T1 = 30