Welcome to the World of Sequences!
Hello future mathematician! This chapter, Sequences, is a fantastic introduction to understanding patterns in numbers. Don't worry if this seems tricky at first—patterns are everywhere in nature and math, and once you spot the trick, the rest is simple!
In this section, you will learn how to describe patterns using rules, find missing numbers, and predict what the 100th number in a list will be without listing them all out. This skill is crucial for later topics in functions and algebra!
Section 1: Understanding What a Sequence Is
What is a Sequence?
In simple terms, a sequence is just an ordered list of numbers that follows a specific rule or pattern.
- Example 1: \(2, 4, 6, 8, 10, \dots\) (The rule is "add 2")
- Example 2: \(1, 3, 9, 27, \dots\) (The rule is "multiply by 3")
Terms and Positions
Each number in the sequence is called a term. We describe where the term is located by its position.
Analogy: A Line of Students
Imagine students lining up for lunch. The first student is in Position 1, the second in Position 2, and so on. In math, we use the letter \(n\) to represent the position number.
| Position (\(n\)) | 1st | 2nd | 3rd | 4th | 5th | ... |
|---|---|---|---|---|---|---|
| Term | \(2\) | \(4\) | \(6\) | \(8\) | \(10\) | ... |
We often call the term in position \(n\) the \(n\)th term.
Key Takeaway: A sequence is an ordered list. We use \(n\) to represent the position number of any term.
Section 2: Generating Sequences Using Rules
There are two main ways we can define the rule for a sequence: the Term-to-Term Rule and the Position-to-Term Rule (\(n\)th term).
1. Term-to-Term Rules (The Simple Rule)
This rule tells you how to get from one term to the very next term in the sequence. You always need to know the starting term to use this rule.
Example: A sequence starts at 5. The rule is "add 3 to the previous term."
- Start: \(5\)
- \(5 + 3 = 8\)
- \(8 + 3 = 11\)
- \(11 + 3 = 14\)
The sequence is: \(5, 8, 11, 14, \dots\)
Did you know? Sequences defined by multiplying the previous term are called Geometric Sequences (e.g., \(2, 6, 18, \dots\)), while those defined by adding or subtracting are called Arithmetic Sequences (our main focus in GCSE).
⚠️ Common Mistake Alert!
When asked to generate the first five terms, ensure you count the starting term! If the first term is given, you only need to calculate four more terms.
Quick Review: Term-to-term rules only tell us how to move one step at a time.
Section 3: The Power of the \(n\)th Term (Position-to-Term Rule)
The Term-to-Term Rule is fine for finding the 5th term, but what if you need the 500th term? Nobody wants to sit there adding 3 four hundred and ninety-nine times!
This is where the Position-to-Term Rule, also known as the \(n\)th term, saves the day!
What is the \(n\)th Term?
The \(n\)th term is an algebraic formula that allows you to calculate any term in the sequence simply by knowing its position (\(n\)).
Analogy: The Vending Machine
Think of the \(n\)th term formula like a vending machine. You put in the position number (your input, \(n\)), and the formula instantly spits out the term (the output).
Example: If the \(n\)th term rule is \(2n + 1\):
- To find the 1st term (\(n=1\)): \(2(1) + 1 = 3\)
- To find the 2nd term (\(n=2\)): \(2(2) + 1 = 5\)
- To find the 100th term (\(n=100\)): \(2(100) + 1 = 201\)
The sequence is \(3, 5, 7, 9, \dots\)
Key Takeaway: The \(n\)th term is a formula that links the position (\(n\)) directly to the value of the term.
Section 4: Focused Study: Arithmetic (Linear) Sequences
Most sequences you will encounter in your International GCSE exam are Arithmetic Sequences (also known as Linear Sequences).
Defining Arithmetic Sequences
An arithmetic sequence is one where the difference between consecutive terms is constant. This constant value is called the common difference.
Example: \(15, 12, 9, 6, \dots\)
- \(12 - 15 = -3\)
- \(9 - 12 = -3\)
- \(6 - 9 = -3\)
The common difference is \(-3\). Since the difference is constant, it is an arithmetic sequence.
Why is this important? In an arithmetic sequence, the common difference is the key number that appears in the \(n\)th term formula!
Think of it this way: If the common difference is +5, the formula must contain \(5n\), because the sequence grows by 5 every time \(n\) increases by 1.
Section 5: Finding the \(n\)th Term for Arithmetic Sequences
This is the most important calculation in this chapter. Follow these three steps every time!
Step-by-Step Guide: Finding the \(n\)th Term (\(Un\))
Let's find the \(n\)th term for the sequence: \(4, 7, 10, 13, 16, \dots\)
Step 1: Find the Common Difference (\(d\))
Calculate how much the sequence increases (or decreases) by between each term.
\(7 - 4 = 3\) \(10 - 7 = 3\)
The common difference (\(d\)) is \(+3\).
This means our \(n\)th term formula will start with \(3n\).
Step 2: Compare to the '3 Times Table'
Write out the sequence we want, and beneath it, write out the sequence generated by our leading term (\(3n\)).
| Position (\(n\)) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Our Sequence | 4 | 7 | 10 | 13 |
| The \(3n\) Sequence (3x table) | \(3 \times 1 = 3\) | \(3 \times 2 = 6\) | \(3 \times 3 = 9\) | \(3 \times 4 = 12\) |
| Difference (What do we add/subtract?) | \(4 - 3 = +1\) | \(7 - 6 = +1\) | \(10 - 9 = +1\) | \(13 - 12 = +1\) |
Step 3: Write the Final Formula
Since we always need to add 1 to the \(3n\) value to get our original sequence, the complete \(n\)th term rule is:
\[ \text{nth Term} = 3n + 1 \]
Memory Aid / Trick: Finding the "Zeroth Term"
A brilliant shortcut is to find the term that would come before the first term (Position \(n=1\)). This is often called the Zeroth Term.
- Start with the sequence: \(\dots, 4, 7, 10, 13, \dots\)
- We know the difference is +3.
- To find the term before 4, we subtract the difference: \(4 - 3 = 1\).
- The Zeroth Term is 1.
- The \(n\)th term is always: (Difference)\(n\) + (Zeroth Term).
- \(n\)th term = \(3n + 1\). (It matches!)
Use this "Zeroth Term" trick for speed and accuracy!
\[ n\text{th Term} = (\text{Common Difference})n + (\text{Zeroth Term}) \]
Section 6: Using the \(n\)th Term
Once you have the formula, you can answer two main types of questions:
1. Finding a Specific Term
If the \(n\)th term of a sequence is \(5n - 8\), find the 20th term.
Method: Replace \(n\) with 20.
\[ \text{20th Term} = 5(20) - 8 \]
\[ 100 - 8 = 92 \]
The 20th term is 92.
2. Checking if a Number is in the Sequence
Is the number 76 a term in the sequence with the rule \(3n + 1\)?
Method: Set the \(n\)th term formula equal to the number, and solve for \(n\). If \(n\) is a whole, positive number, the number is in the sequence.
Step 1: Set up the equation.
\[ 3n + 1 = 76 \]
Step 2: Solve for \(n\).
Subtract 1 from both sides:
\[ 3n = 75 \]
Divide by 3:
\[ n = \frac{75}{3} \]
\[ n = 25 \]
Since \(n=25\) (a whole number), Yes, 76 is the 25th term in the sequence.
⚠️ What if the number is NOT in the sequence?
If you solved for \(n\) and got a fraction or a decimal (e.g., \(n = 25.5\)), it means that the number lies somewhere between two terms, and therefore is not part of the sequence.
Key Takeaway: The \(n\)th term allows you to quickly find any term or check if a number belongs by testing if its position \(n\) is a positive integer.
Summary Checklist: Sequences
- Sequence: An ordered list of numbers following a rule.
- Term: A number in the sequence.
- Position (\(n\)): Where the term is located.
- Arithmetic Sequence: Has a constant common difference (\(d\)).
- \(n\)th Term Formula: The rule that allows you to calculate a term based on its position. Use the "Difference and Zeroth Term" method to find it quickly!
Keep practising your algebra skills alongside this chapter, as solving for \(n\) is crucial for success!