# Number Patterns and Sequences

Updated: Apr 18, 2020

Number Patterns is one of the chapters in both N-level** **and O-Level E-Math where students rarely have 100% confidence of getting it right during the exams. Most students are pretty **good at identifying patterns** and have no problem spotting the logic behind each sequence. However, the difficulty often lies in **coming up with a formula** or **an equation** that expresses the **nth-term** of the number sequence **in terms of n.**

Since **every pattern is different**, some students do not rely on formulas and they depend purely on their superior understanding to construct the nth term formula every time. However, this is relatively **time consuming** and the time spent is often not worth the marks allocated.

In this article, we will describe **two** of the **more common patterns** and introduce their **general formulas** such that there is a more **structured** and **efficient** way of deriving the **nth-term formula.**

__Number Pattern Type 1: Constant Difference__

Find the formula of the **nth** **term** of the sequence: 12, 15, 18, 21, 24, ...

Notice that every term is **3 more** than the **previous** term. This is the **simplest** **kind** of pattern where the **increase/decrease** in **constant**.

The **General** **formula** of this **nth** **term** for this number pattern sequence is:

**n-th term = a + d (n - 1)**

where **d** is the difference between the terms,

**a** is the first term

and **n** is the term number

Let’s look back at the **above** **example**:

12, 15, 18, 21, 24, …

d = 3

a = 12

**n-th** term = 12 + 3 (n – 1)

= 12 + 3n – 3

= 9 + 3n

__Number Pattern Type 2: Increasing Difference__

** We call this a We call this a **pattern in a pattern**

Find the formula of the **nth** term of the sequence: 8, 13, 20, 29, 40, …

Notice that the **difference** **between the terms** is **increasing / decreasing** at a **constant** rate.

The General formula of this nth term for this number pattern sequence is:

**n-th** term = a + d (n – 1) + c/2 (n – 1) (n – 2)

where **d** is the first **difference**,

**a** is the first** term**,

**c** is the **difference of the difference**

and **n** is the term number

Let’s look back at the **above example**:

8, 13, 20, 29, 40, …

a = 8, d = 5, c = 2

**n-th** term = 8 + 5 (n – 1) + 2/2 (n – 1) (n – 2)

= 8 + 5n – 5 + n^2 – n – 2n + 2

= n^2 + 2n + 5

Think you have understood the entire topic? Let’s try out the practice questions below!

__Questions__

Find the nth-term formula for the following sequences:

a) 5, 9, 13, 17, …

b) -7, -4, -1, 2, …

c) 6, 4, 2, 0, -2, …

d) 3, 5, 8, 12, 17, …

e) 0, 3, 8, 15, 24, …

__Answers__

a) nth term = 4n+1

b) nth term = 3n-10

c) nth term = -2n+8

d) nth term = 0.5 (n^2+n+4)

e) nth term = n^2 - 1