Lesson 04: Number Patterns _ Part 02 The general term of a number pattern

Before we proceed with the analysis of sequences, we need to establish a convenient way of writing the elements of a sequence. In this section, we will introduce a notation that will help us express the general term of a sequence and its properties.

Figure 01

For example, let us be familiar with the notation as mentioned in the above Figure 01 to the below given sequence. ( written up to 3 terms only)

5, 11, 17, 23, …

A sequence can have a pattern or a rule that determines how to get the next term as aforementioned details in Lesson 03. The nth term of a sequence is the term that corresponds to the position n in the list. When considering a value n, the corresponding nth term is denoted as following;

This is called the general term of the sequence.

Now let us focus on one of the common tasks in Mathematics to find the nth term of a sequence. This means that we want to know the value of the term that corresponds to a given positive integer n, such as 1, 2, 3, and so on.

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Suppose we want to find the 80th term of the sequence 5, 11, 17, 23…, which is a sequence with a common difference. Below are the few steps to generate the general term and find the requested answer.

1. Find the common difference of the sequence 5, 11, 17, 23…
   common difference = any term except the first term — the previous term
   common difference = 17–11 = 6
2. Think a method where we can recreate the common difference into first term “5”.
   first term = 6 x 1 — 1 = 5
3. Think a method where we can recreate the common difference into second term “11”.
   second term = 6 x 2 — 1 = 11
4. Think a method where we can recreate the common difference into third term “17”.
   third term = 6 x 3 — 1 = 17
5. Finally, think a method where we can recreate the common difference into nth term Tn.
   nth term Tn = 6 x n — 1 = 6n — 1
6. Accordingly, by t he obtained Tn (general term) we can obtain, 80th term of the sequence 5, 11, 17, 23….
   T80 = 6 x 80 — 1 = 479

To test your comprehension of the concept of finding the general term with the common difference, you can practice the following exercise on your own. It will help you to verify if you can apply the formula and identify the common difference in different sequences. The exercise is not mandatory, but we encourage you to try it for your own benefit.

Find the common difference in each sequence.