Sequences from differences, powers and ratios
Connect a pattern to its position rule, rather than confusing the next-step rule with the nth term.
Core: continuing patterns and nth terms for linear, simple quadratic and simple cubic sequences. Extended adds more general polynomial, exponential and combined sequences.
Before you begin
- Substitute an integer into an expression.
- Calculate squares, cubes and signed differences.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the value of n² + 1 at n = 3?
What you will learn
- Find and use linear, simple quadratic and simple cubic nth terms.
- Distinguish position rules from term-to-term rules.
- Recognise exponential sequences and combinations in Extended work.
A linear rule uses the position
The sequence 5, 8, 11, 14 has common difference three. Its nth term is 3n + 2 for n = 1, 2, 3, … because 3n gives 3, 6, 9, 12 and the constant adds two to each. Adding three describes how to move from one term to the next; it does not directly locate the hundredth term.
For a first term a and constant difference d, Tₙ = a + (n − 1)d. The factor n − 1 counts the steps after the first term. A decreasing sequence uses a negative d. To test membership, set the nth term equal to the proposed value and check that n is a positive integer.
Compare with squares and cubes
The sequence 2, 5, 10, 17 follows n² + 1. Its first differences are 3, 5, 7 and its second differences are all two. In general, a quadratic rule an² + bn + c has constant second difference 2a. After finding a, subtract an² from each term to expose a linear remainder.
The simple cubic sequence 3, 10, 29, 66 matches n³ + 2. Cubic sequences have constant third differences; for an³ + bn² + cn + d that constant is 6a. Core problems use simple patterns; more involved coefficients and combinations are Extended. Always verify a proposed rule against all stated terms.
Extended: repeated ratios and combined patterns
The sequence 3, 6, 12, 24 multiplies by two each step, so Tₙ = 3 × 2ⁿ⁻¹. The exponent is n − 1 because the first term has undergone no doubling. A constant ratio indicates an exponential pattern; a constant difference indicates a linear one.
A pattern such as 3, 6, 11, 20 can be recognised as 2ⁿ + n for positions one through four. For a general quadratic such as 6, 17, 34, 57, the constant second difference is six, so the n² coefficient is three. Subtracting 3n² leaves 3, 5, 7, 9, whose rule is 2n + 1. The full rule is 3n² + 2n + 1. A finite list can fit more than one invented rule; justify the intended simple structure rather than claiming a unique infinite sequence.
Pause and explain
For 5, 8, 11, …, which expression is the nth term?
Worked example
Find the nth term of 4, 9, 16, 25, … and its twentieth term.
Show the worked solution
- These are 2², 3², 4² and 5², so the square's input is one more than the position.
- Tₙ = (n + 1)² = n² + 2n + 1; test it at n = 1 and n = 4.
- T₂₀ = 21² = 441.
Answer Tₙ = (n + 1)²; T₂₀ = 441
From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the nth term of 7, 11, 15, 19, … and its tenth term.
Give me a hint
The common difference supplies the coefficient of n.
Compare my reasoning
- The difference is four, so start with 4n.
- At n = 1, 4n is four; add three to match seven.
- Tₙ = 4n + 3 and T₁₀ = 43.
Tₙ = 4n + 3; T₁₀ = 43
Look for these in your work
- Used position n rather than a term's value.
- Checked the first term.
Find simple nth-term rules for 4, 7, 12, 19 and for 0, 7, 26, 63.
Give me a hint
Compare the terms with n² and n³ respectively.
Compare my reasoning
- Subtract n² from the first sequence to leave the constant three.
- Subtract n³ from the second sequence to leave the constant negative one.
- The rules are n² + 3 and n³ − 1; each reproduces all four given terms.
n² + 3; n³ − 1
Look for these in your work
- Identified the underlying power pattern.
- Verified more than the first term.
Extended: a pattern uses 5 tiles in stage one and triples at every next stage. Write the position rule and determine the first stage with more than 400 tiles.
Give me a hint
The first stage corresponds to exponent zero.
Compare my reasoning
- Tₙ = 5 × 3ⁿ⁻¹.
- Stages four and five have 135 and 405 tiles respectively.
- Since each stage increases, stage five is the first above 400.
Tₙ = 5 × 3ⁿ⁻¹; stage 5
Look for these in your work
- Used a ratio model with the correct starting exponent.
- Checked the preceding stage against the threshold.
Extended: find a cubic rule for 3, 13, 37, 81, …, given that it combines n³ with a quadratic and a constant.
Give me a hint
Subtract n³ from the terms and identify the simpler remainder.
Compare my reasoning
- First differences are 10, 24, 44; second differences are 14, 20, so the third difference is six and the cubic coefficient is one.
- Subtracting 1, 8, 27, 64 leaves 2, 5, 10, 17, the sequence n² + 1.
- The rule is n³ + n² + 1; it reproduces all four terms and predicts 151 at n = 5.
Tₙ = n³ + n² + 1; T₅ = 151
Look for these in your work
- Used differences to confirm the leading degree.
- Recovered the remainder rule and checked the combined expression.
Common mistakes
- Writing a term-to-term change as an nth-term formula.
- Starting an exponential power at n instead of n − 1 without adjusting its coefficient.
- Checking a rule against only one term.
What can you explain now?
For Tₙ = 3n + 2, is 50 a term?
Compare with the explanation
Yes, the 16th term
3n + 2 = 50 gives n = 16, a positive integer.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Build linear, quadratic and cubic sequences from rules of your own. Recover each rule from its terms. Extended: also build a geometric sequence.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons and a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other six syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice is self-checked, not automatically graded. The checkpoint samples skills and does not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources