Introduction to Sequences
What are sequences?
- A sequence is an ordered set of numbers that follow a rule
- For example 3, 6, 9, 12...
- The rule is to add 3 each time
- For example 3, 6, 9, 12...
- Each number in a sequence is called a term
- The location of a term within a sequence is called its position
- The letter n is used for position
- n = 1 refers to the 1st term
- n = 2 refers to the 2nd term
- If you do not know its position, you can say the n th term
- The letter n is used for position
- Another way to show the position of a term is using subscripts
- A general sequence is given by a1, a2, a3, ...
- a1 represents the 1st term
- a2 represents the 2nd term
- an represents the nth term
- A general sequence is given by a1, a2, a3, ...
How do I write out a sequence using a term-to-term rule?
- Term-to-term rules tell you how to get the next term from the term you are on
- It is what you do each time
- For example, starting on 4, add 10 each time
- 4, 14, 24, 34, ...
How do I write out a sequence using a position-to-term rule?
- A position-to-term rule is an algebraic expression in n that lets you find any term in the sequence
- This is also called the n th term formula
- You need to know what position in the sequence you are looking for
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- To get the 1st term, substitute in n = 1
- To get the 2nd term, substitute in n = 2
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- You can jump straight to the 100th term by substituting in n = 100
- You do not need to find all 99 previous terms
- For example, the n th term is 8n + 2
- The 1st term is 8×1 + 2 = 10
- The 2nd term is 8×2 + 2 = 18
- The 100th term is 8×100 + 2 = 802
How do I know if a value belongs to a sequence?
- If you know the n th term formula, set the value equal to the formula
- This creates an equation to solve for n
- For example, a sequence has the n th term formula 8n + 2
- Is 98 in the sequence?
- It is in the sequence, it is the 12th term
- Is 124 in the sequence?
- n is not a whole number, so it is not in the sequence
- Is 98 in the sequence?
Examiner Tip
- In the exam, it helps to write the position number (the value of n) above each term in the sequence.
Worked example
A sequence has the n th term formula 3n + 2.
Substitute n = 1, n = 2 and n = 3 into the formula
3×1 + 2 = 5
3×2 + 2 = 8
3×3 + 2 = 11
5, 8, 11
Substitute n = 80 into the formula
3×80 + 2
The 80th term is 242
Set the formula equal to 96
Solve to find n
If n is a whole number, it is a term in the sequence
94 is not divisible by 3
The nearest possible value is 95 ((95-2) ÷ 3 = 31, the 31st term)
96 is not in the sequence