Edexcel GCSE Maths: Sequences and Patterns

Edexcel GCSE Maths: Sequences and Patterns

Introduction

Sequences are lists of numbers that follow a specific rule. Understanding sequences is crucial in mathematics, as they can be used to model real-life situations and predict future events. This tutorial will introduce you to two common types of sequences: arithmetic and geometric.

Arithmetic Sequences

An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference.

Example: 2, 5, 8, 11, 14...

In this sequence, the common difference is 3.

Finding the nth term:

The nth term of an arithmetic sequence is given by the formula:

an = a1 + (n - 1)d

where:

• an is the nth term
• a1 is the first term
• d is the common difference
• n is the position of the term in the sequence

Example: Find the 10th term of the sequence 2, 5, 8, 11, 14...

Here, a1 = 2, d = 3, and n = 10.

Therefore, the 10th term is:

a10 = 2 + (10 - 1)3 = 29

Geometric Sequences

A geometric sequence is a sequence where the ratio between consecutive terms is constant. This constant ratio is called the common ratio.

Example: 2, 6, 18, 54...

In this sequence, the common ratio is 3.

Finding the nth term:

The nth term of a geometric sequence is given by the formula:

an = a1 * r^(n-1)

where:

• an is the nth term
• a1 is the first term
• r is the common ratio
• n is the position of the term in the sequence

Example: Find the 6th term of the sequence 2, 6, 18, 54...

Here, a1 = 2, r = 3, and n = 6.

Therefore, the 6th term is:

a6 = 2 * 3^(6-1) = 486

Real-Life Applications

Sequences are used to model various real-life situations, such as:

• Population growth: The population of a city can be modeled using an arithmetic or geometric sequence.
• Compound interest: The amount of money in a savings account can be modeled using a geometric sequence.
• Depreciation: The value of a car can be modeled using a geometric sequence.

Solving Pattern Problems

Sequences are often used in pattern problems. To solve these problems, you need to identify the type of sequence and use the appropriate formula to find the missing term.

Example:

A pattern of tiles starts with 3 tiles in the first row. Each subsequent row has 2 more tiles than the previous row. How many tiles are in the 7th row?

This is an arithmetic sequence with a1 = 3 and d = 2. We need to find a7.

Using the formula:

a7 = 3 + (7 - 1)2 = 15

Therefore, there are 15 tiles in the 7th row.

Conclusion

Understanding sequences is a fundamental skill in GCSE Maths. By learning the concepts of arithmetic and geometric sequences, you can model real-life situations, solve pattern problems, and predict future events. Remember to practice regularly to solidify your understanding.