Sequences
Section: Algebra | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
What is a Sequence?
A sequence is an ordered list of numbers, each called a term, that follows a rule.
- Term-to-term rule: describes how to get from one term to the next, e.g. "add 5".
- Position: where a term sits in the sequence, given by n (1st term is n=1, 2nd term is n=2, and so on).
- nth term (position-to-term rule): a formula that gives any term directly from its position, n, without needing the term before it.
Generating Sequences from a Term-to-Term Rule
Given a first term and a rule, each new term is generated by applying the rule to the term before it.
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Question: Generate the first 4 terms starting at 3, with the rule "add 5".
- Step 1: 1st term = 3
- Step 2: 2nd term = 3 + 5 = 8. 3rd term = 8 + 5 = 13. 4th term = 13 + 5 = 18
- Answer: 3, 8, 13, 18
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Question: Generate the first 4 terms starting at 40, with the rule "subtract 7".
- Step 1: 1st term = 40
- Step 2: 2nd term = 40 - 7 = 33. 3rd term = 33 - 7 = 26. 4th term = 26 - 7 = 19
- Answer: 40, 33, 26, 19
Term-to-Term Rules Using Indices
A term-to-term rule isn't always a simple add or subtract - it can also involve squaring, cubing, or a decimal step.
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Question: The first term of a sequence is 2. The term-to-term rule is "square the term, then subtract 1". Find the first three terms.
- Step 1: 1st term = 2
- Step 2: 2nd term = 2² − 1 = 3
- Step 3: 3rd term = 3² − 1 = 8
- Answer: 2, 3, 8
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Question: The first term of a sequence is 4, with the rule "add 1.5". Find the first four terms.
- Step 1: 1st term = 4
- Step 2: 2nd term = 4 + 1.5 = 5.5. 3rd term = 5.5 + 1.5 = 7. 4th term = 7 + 1.5 = 8.5
- Answer: 4, 5.5, 7, 8.5
Generating a Sequence from a Growing Pattern
Sequences often come from a picture that grows in a regular way. Count how many items are in each pattern first, then treat those counts as an ordinary number sequence.
Pattern n is an n-by-n grid of tiles, plus a fixed row of 4 header tiles
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Question: The diagram shows the first three patterns in a tile design. Pattern n is an n-by-n grid of tiles, with 4 extra tiles fixed above it. Find the number of tiles in each of the first four patterns, then find the nth term.
- Step 1: Count each pattern: Pattern 1 = 1 + 4 = 5. Pattern 2 = 4 + 4 = 8. Pattern 3 = 9 + 4 = 13. Pattern 4 = 16 + 4 = 20
- Step 2: Sequence is 5, 8, 13, 20. First differences: 3, 5, 7 - not constant, so check second differences: 2, 2 - constant, so the sequence is quadratic
- Step 3: Compare to the square numbers 1, 4, 9, 16: each term is exactly 4 more
- Answer: 5, 8, 13, 20; nth term = n² + 4
Linear vs Quadratic Sequences
Sequences can also come from numerical or visual patterns. Looking at the differences between consecutive terms reveals what type of sequence it is. A sequence without a constant difference is called non-linear; the most common non-linear sequences at this stage are quadratic, where the second differences are constant.
A dot pattern for the square numbers 1, 4, 9, 16 - a classic quadratic sequence
| Type | How to Identify | Example |
|---|---|---|
| Linear | 1st differences are constant | 3, 7, 11, 15 (always +4) |
| Quadratic | 1st differences change, but 2nd differences are constant | 1, 4, 9, 16, 25 (1st diffs: 3, 5, 7, 9; 2nd diff always +2) |
Finding the nth Term of a Linear Sequence
A linear sequence has an nth term of the form an ± b, where a is the common difference. Find a first, then adjust with b so the formula matches the first term.
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Question: Find the nth term of 5, 8, 11, 14, 17.
- Step 1: Common difference = 3, so start with 3n
- Step 2: When n = 1, 3n = 3, but the actual term is 5 - a difference of +2
- Answer: nth term = 3n + 2 (check: n=5 gives 3×5+2=17 ✓)
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Question: Find the nth term of 40, 33, 26, 19.
- Step 1: Common difference = -7, so start with -7n
- Step 2: When n = 1, -7n = -7, but the actual term is 40 - a difference of +47
- Answer: nth term = -7n + 47 (check: n=4 gives -28+47=19 ✓)
Other nth Term Forms
Not every sequence is linear. The nth term can also take the form n/a, n², n³, or n² ± a, where a is a whole number.
| Form | Example Sequence | nth Term |
|---|---|---|
| n² | 1, 4, 9, 16, 25 | n² |
| n² ± a | 4, 7, 12, 19, 28 | n² + 3 |
| n³ | 1, 8, 27, 64, 125 | n³ |
| n/a | 1/2, 1, 3/2, 2, 5/2 | n/2 |
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Question: Find the nth term of 4, 7, 12, 19, 28.
- Step 1: Compare to the square numbers 1, 4, 9, 16, 25
- Step 2: Each term is exactly 3 more: 4-1=3, 7-4=3, 12-9=3, 19-16=3, 28-25=3
- Answer: nth term = n² + 3
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Question: Find the nth term of 1/2, 1, 3/2, 2, 5/2.
- Step 1: Multiply every term by 2: 1, 2, 3, 4, 5 - this matches n exactly
- Step 2: Since doubling the sequence gives n, the original sequence is n ÷ 2
- Answer: nth term = n/2
Checking Whether a Number is a Term in a Sequence
Given an nth term rule, you can check whether a particular number belongs to the sequence: set the nth term expression equal to that number and solve for n. If n comes out as a positive whole number, the value is a term; if not, it isn't.
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Question: A sequence has nth term n² + 5. Is 86 a term in this sequence?
- Step 1: Set n² + 5 = 86, so n² = 81
- Step 2: n = 9, which is a positive whole number
- Answer: Yes - 86 is the 9th term
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Question: A sequence has nth term n² + 5. Is 90 a term in this sequence?
- Step 1: Set n² + 5 = 90, so n² = 85
- Step 2: n = √85 ≈ 9.22, which is not a whole number
- Answer: No - 90 is not a term in this sequence
Real-World Applications
Sequences describe many growing patterns:
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