Understanding Number Patterns: A Step-by-Step Guide

In this lesson, we'll delve into the concept of number patterns, focusing specifically on how to find the next term in a sequence and how to determine the nth term formula. This is a fundamental skill in mathematics, particularly for those preparing for the IGCSE exams.

Recognizing the Pattern

Let's start with the sequence provided: 22, 17, 12, 7, 2. The first task is to find the next term in this sequence. A good approach here is to observe the changes between consecutive terms. Notice how each term decreases by 5:

• From 22 to 17, we subtract 5.
• From 17 to 12, we subtract 5.
• From 12 to 7, we subtract 5.
• From 7 to 2, we subtract 5.

Since the sequence consistently decreases by 5, the next term would be 2 - 5, which equals -3. So, the next term in the sequence is -3.

Defining the nth Term

The nth term of a sequence is a general formula that allows us to find any term in the sequence without having to list all the terms. For example, in our sequence, we want to find a formula that can directly give us any term, depending on its position (n) in the sequence.

The nth term is represented by T_n. In our case, T₁ = 22, T₂ = 17, T₃ = 12, and so on. The sequence decreases by the same amount each time, which suggests a linear relationship, meaning the sequence can be represented by a straight line if plotted on a graph.

Finding the nth Term Formula

To find the nth term formula, we need to determine two things: the common difference (which we've identified as -5) and the starting value (which we'll calculate next).

We can start by using the linear equation:

y = mx + c

In the context of sequences, we adapt this to:

T_n = an + q

Where:

• a is the common difference (in our case, -5)
• n is the position of the term in the sequence
• q is a constant we need to find

We already know that a = -5. Now, let's find q by substituting a known term into the equation. Let's use T₃ (which is 12):

12 = -5(3) + q

Simplifying:

12 = -15 + q

To solve for q:

q = 12 + 15 = 27

So, our nth term formula becomes:

T_n = -5n + 27

Conclusion

By following these steps, we’ve derived the formula for the nth term of our sequence: T_n = -5n + 27. This formula allows us to find any term in the sequence. For example, if you wanted to find the 6th term, you would simply substitute n = 6 into the formula:

T₆ = -5(6) + 27 = -30 + 27 = -3

This confirms our earlier calculation. Understanding how to derive and use the nth term formula is crucial in mastering number patterns, a key topic in your IGCSE Maths exams. Keep practicing, and these concepts will become second nature.