🎯 Introduction to Sequences
What Are Sequences?
A sequence is an ordered list of numbers that follow a specific rule or pattern. Each number in the sequence is called a term, and its location within the sequence is called its position.
📌 Example: The Sequence 3, 6, 9, 12, ...
Each term is found by adding 3 to the previous term. This "add 3 each time" is the rule.
Key Idea: We use n to represent position. So n = 1 means the 1st term, n = 2 means the 2nd term, and so on. If we don't know the position, we call it the nth term.
Writing Sequences with Notation
A general sequence is written as: a₁, a₂, a₃, ..., aₙ
- a₁ = the 1st term
- a₂ = the 2nd term
- aₙ = the nth term (any position you choose)
Two Ways to Describe Sequences
1. Term-to-Term Rules
This tells you how to get the next term from the current term. You start with a first term and apply the rule repeatedly.
📌 Example: Starting on 4, add 10 each time
4 → 14 → 24 → 34 → 44 → ...
Q1: If a sequence starts with 2 and the term-to-term rule is "multiply by 3 each time", what are the first four terms?
2. Position-to-Term Rules (The nth Term Formula)
This is an algebraic formula in terms of n that lets you find any term directly without calculating all the previous ones. This is powerful because you can jump straight to the 100th term if you want.
📌 Example: nth term = 8n + 2
n = 1: 8(1) + 2 = 10 (1st term)
n = 2: 8(2) + 2 = 18 (2nd term)
n = 100: 8(100) + 2 = 802 (100th term)
The Power: With a position-to-term rule, you never have to calculate all 99 previous terms. Just substitute n and you're done.
Testing if a Number Belongs to a Sequence
If you know the nth term formula, you can check whether a specific number is in the sequence by setting it equal to the formula and solving for n. If n is a positive whole number, it's in the sequence. If n is a fraction or negative, it's not.
📌 Example: Is 98 in the sequence with nth term = 8n + 2?
Set 8n + 2 = 98
8n = 96
n = 12
Since n = 12 (a whole number), yes, 98 is the 12th term.
Q2: Is 124 in the sequence with nth term = 8n + 2? Explain why or why not.
❌ Common Mistake
Forgetting to check that n is a whole number. A value like n = 15.25 means the number is not in the sequence, even if it seems close to a term.
📈 Linear Sequences (Arithmetic Sequences)
What Is a Linear Sequence?
A linear sequence (also called an arithmetic sequence) goes up or down by the same amount every time. This constant amount is called the common difference, written as d.
📌 Examples of Linear Sequences
Finding the nth Term Formula for a Linear Sequence
The formula is: nth term = dn + b
Where:
- d = the common difference (what you add each time)
- b
before