Introduction to Sequences

Welcome to the world of sequences! If you have ever looked at a pattern of numbers and wondered what comes next, you are already doing sequence math. In this chapter, we will learn how to spot patterns, find rules, and even predict numbers that are way far down the line. Learning sequences is like learning the "DNA" of a pattern. Whether it’s how a sunflower grows or how a computer calculates data, sequences are everywhere. Don’t worry if this seems a bit abstract at first—once you see the patterns, it’s like solving a puzzle!

1. The Basics: What is a Sequence?

A sequence is simply a list of numbers in a specific order. Each number in the sequence is called a term. To understand a sequence, we usually look for a rule. There are two main ways to describe these rules:

Term-to-term Rule

This rule tells you how to get from one number to the very next one. Example: 5, 8, 11, 14... The term-to-term rule here is "add 3".

Position-to-term Rule (\(n\)th term)

This is a more powerful rule. It uses the letter \(n\) to represent the position of the number in the list. - If \(n = 1\), it’s the 1st term. - If \(n = 10\), it’s the 10th term. Using a position-to-term rule allows you to find the 100th term without having to write out all the numbers in between! Quick Review: - Term: A number in the sequence. - \(n\): The position (1st, 2nd, 3rd...). - Term-to-term: Just the next step. - Position-to-term: A formula to find any term.

2. Special Sequences You Need to Know

The AQA syllabus expects you to recognise a few famous sequences. Think of these as the "celebrities" of the math world!

Square and Cube Numbers

- Square Numbers: \(1, 4, 9, 16, 25...\) (Rule: \(n^2\)) - Cube Numbers: \(1, 8, 27, 64, 125...\) (Rule: \(n^3\))

Triangular Numbers

These are numbers that can form an equilateral triangle. - Sequence: \(1, 3, 6, 10, 15...\) Memory Aid: Imagine stacking bowling pins. 1 on top, 2 below it, 3 below that. The totals are your triangular numbers!

Fibonacci-type Sequences

In these sequences, you find the next term by adding the two previous terms together. Example: 1, 1, 2, 3, 5, 8, 13... (because \(1+1=2\), \(1+2=3\), \(2+3=5\), etc.) Did you know? Fibonacci sequences appear constantly in nature, from the petals on a flower to the spiral of a galaxy!

Arithmetic and Geometric Progressions

- Arithmetic Progression: You add or subtract the same number every time. (e.g., \(10, 7, 4, 1...\) - here we subtract 3). - Geometric Progression: You multiply or divide by the same number every time. (e.g., \(3, 6, 12, 24...\) - here we multiply by 2). Key Takeaway: Always check if the sequence is adding (Arithmetic) or multiplying (Geometric) first!

3. Finding the \(n\)th Term of a Linear Sequence

A linear sequence (or Arithmetic Progression) is one where the "gap" between numbers is always the same. Finding the \(n\)th term formula looks like this: \(dn + c\).

Step-by-Step: Finding the Rule

Let's find the \(n\)th term for: 7, 12, 17, 22... 1. Find the Difference (\(d\)): What is the gap? Between 7 and 12, the gap is +5. So, our rule starts with \(5n\). 2. Find the "Zero-th" Term (\(c\)): Imagine there was a number before the first term. If we go backward from 7 by subtracting 5, we get 2. 3. Put it together: The \(n\)th term is \(5n + 2\).

How to Check Your Answer

If you want the 1st term, plug in \(n = 1\): \(5(1) + 2 = 7\). It works! Common Mistake: Students often forget the sign of the difference. If the sequence is going down (e.g., \(20, 17, 14...\)), the difference is negative (e.g., \(-3n\)).

4. Quadratic Sequences (Higher Tier Only)

Sometimes, the difference between numbers isn't constant, but the difference of the differences is! These are Quadratic Sequences. They always follow the format: \(an^2 + bn + c\)

How to Spot a Quadratic Sequence

Look at: 2, 6, 12, 20... - First differences: 4, 6, 8 - Second differences: 2, 2 Because the second difference is constant, it is quadratic. Step-by-step Tip: The value of \(a\) is always half of the second difference. In the example above, the second difference is 2, so \(a = 1\). The sequence starts with \(1n^2\). Key Takeaway: If the first gap changes, find the second gap!

Summary and Tips for Success

- Read the question carefully: Does it ask for the next term (simple addition) or the \(n\)th term (formula)? - Check your work: Always plug \(n=1\) and \(n=2\) back into your formula to see if you get the first two numbers of the sequence. - Don't Panic: If a sequence looks weird, write out the differences between the numbers. Often, the pattern will reveal itself once you see the gaps! Quick Review Box: - Linear: \(dn + c\) (constant first difference) - Quadratic: \(an^2 + bn + c\) (constant second difference) - Geometric: Multiply/Divide by a "common ratio" - Fibonacci: Add the two previous terms