Learning Objectives
  • Continue sequences using term-to-term rules and structural patterns.
  • Find nth terms of linear sequences.
  • Identify quadratic, cubic and exponential sequences from differences or ratios.
  • Use nth-term rules to find specified terms and test membership.
  • Describe relationships between related sequences and visual patterns.
Key Terms
Sequence
An ordered list of numbers or objects following a rule.
Term
A single member of a sequence.
Term-to-term rule
A rule that generates each term from the previous term.
Nth term
A formula giving a term directly from its position n.
First difference
The difference between consecutive terms.
Second difference
The difference between consecutive first differences.
Exponential sequence
A sequence with a constant multiplicative ratio.
8th Edition Chapter Map
  • Term-to-term rules
  • Linear nth terms
  • Quadratic and cubic patterns
  • Exponential sequences
  • Visual patterns and membership tests
Term-To-Term Rules

A term-to-term rule explains how to move from one term to the next. It may involve adding, subtracting, multiplying, dividing or alternating operations. It is useful for continuing a sequence but not always efficient for finding a distant term.

Worked Example: Continue A Sequence

Question: Find the next three terms of 5,9,13,17,\ldots.

  1. The first difference is consistently +4.
  2. Add 4 repeatedly.

Answer: 21,25,29.

Linear Sequences

A linear sequence has constant first difference d. Its nth term has the form dn+c. Find c by comparing dn with the actual sequence.

Worked Example: Find A Linear Nth Term

Question: Find the nth term of 7,12,17,22,\ldots.

  1. The common difference is 5, so begin with 5n.
  2. 5n gives 5,10,15,20.
  3. Add 2 to reach the given sequence.

Answer: 5n+2.

Using An Nth Term
Worked Example: Membership

Question: Is 152 a term of the sequence with nth term 7n-2?

  1. Set 7n-2=152.
  2. 7n=154, so n=22.
  3. Because 22 is a positive integer, 152 is a term.

Answer: Yes, it is the 22nd term.

Quadratic Sequences

A quadratic sequence has constant second difference. If the second difference is 2a, the nth term begins with an^2. Subtract this basic quadratic sequence and find the remaining linear expression.

Worked Example: Quadratic Nth Term

Question: Find the nth term of 3,8,15,24,35,\ldots.

  1. First differences: 5,7,9,11. Second differences: 2,2,2.
  2. So a=1 and begin with n^2.
  3. n^2 gives 1,4,9,16,25. The difference from the sequence is 2,4,6,8,10=2n.

Answer: n^2+2n.

Cubic Sequences

A cubic sequence has constant third differences. The Cambridge syllabus can include cubic sequences and simple combinations. At this level, patterns often relate to known cubes such as n^3 with an added linear or constant expression.

Worked Example: Recognise A Cubic Pattern

Question: Find a rule for 2,10,30,68,130,\ldots.

  1. Compare with n^3: 1,8,27,64,125.
  2. Each given term is 1, 2, 3, 4, 5 more respectively.
  3. The added part is n.

Answer: n^3+n.

Exponential Sequences

An exponential sequence has a constant ratio rather than a constant difference. For example, 3,6,12,24,\ldots multiplies by 2. A suitable nth term is 3\times2^{n-1}.

Sequence type Diagnostic feature Typical nth term
Linear Constant first difference an+b
Quadratic Constant second difference an^2+bn+c
Cubic Constant third difference an^3+bn^2+cn+d
Exponential Constant ratio ar^{n-1}
Alternating And Combined Patterns

Some sequences alternate signs or combine two simple sequences. For 2,-4,6,-8,10,\ldots, the magnitude follows 2n while the sign alternates. You may be asked only to continue or describe such a pattern rather than form an advanced formula.

Visual Patterns

For matchstick or tile patterns, identify what is present in the first figure and what is added for each new figure. An expression such as 4+3(n-1) can then be simplified to 3n+1. Explain what each part counts.

Examination Guidance
  • Write a difference table neatly for quadratic or cubic sequences.
  • When using an nth term, remember that n must be a positive integer for sequence membership.
  • Check a proposed nth term against at least the first three terms.
  • Distinguish a constant difference from a constant ratio.
Common Mistakes
  • Assuming every sequence with increasing differences is quadratic without checking second differences.
  • Using ar^n instead of ar^{n-1} when a is the first term.
  • Forgetting that the first term corresponds to n=1.
  • Giving only a term-to-term rule when an nth term is requested.
Knowledge Check And Practice

1. Find the nth term of 4,9,14,19,\ldots.

Answer: 5n-1.

2. Find the 40th term of 3n+7.

Answer: 127.

3. Is 100 a term of 6n+1?

Answer: No. 6n+1=100 gives n=16.5, not an integer.

4. Find the nth term of 2,6,12,20,30,\ldots.

Answer: n^2+n.

5. Find the next two terms of 5,15,45,135,\ldots.

Answer: 405,1215.
Extended Worked Practice
Finding A Quadratic Nth Term

Question: Find the nth term of 4,11,22,37,56,\ldots.

  1. First differences are 7, 11, 15, 19; second differences are constant at 4.
  2. For an^2, the second difference is 2a, so 2a=4 and a=2.
  3. Subtract 2n^2 from the sequence: 2, 3, 4, 5, 6, giving n+1.

Answer: 2n^2+n+1.

Using An Nth-Term Formula

Question: The nth term is 5n-8. Find the 40th term and determine whether 117 is in the sequence.

  1. 40th term: 5(40)-8=192.
  2. Set 5n-8=117, so 5n=125 and n=25.

Answer: The 40th term is 192; 117 is the 25th term.

Building A Tile Pattern Formula

Question: Figure 1 uses 7 tiles, figure 2 uses 11 tiles and figure 3 uses 15 tiles. Assume the pattern continues linearly. Find the formula and the number of tiles in figure 50.

  1. The common difference is 4, so start with 4n.
  2. At n=1, 4n=4, which is 3 too small.

Answer: T_n=4n+3, hence T_{50}=203.

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