Sequences from patterns – difference tables, the nth term and recurrence relations
MathematicsSequences & Financial MathAges 16–17
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Sign in to playGrow matchstick, tile and cube patterns, record the terms in a table and plot them. Use first, second and third differences to find the nth term of linear (an + b), quadratic and cubic sequences, check whether a number is a term, and compare with geometric and Fibonacci-type sequences. The Recurrence screen iterates uₙ₊₁ = f(uₙ) on a cobweb diagram to show increasing, decreasing, convergent and periodic sequences, and adds the first N terms in sigma notation.
Lesson: Sequences: the nth term from patterns, the method of differences, recurrence relations, increasing, decreasing and periodic sequences, sigma notation
What it shows
A sequence can be described by a position-to-term rule (the nth term uₙ) or by a term-to-term rule (a recurrence relation). The method of differences finds the nth term: constant first differences give a linear sequence an + b, constant second differences k give a quadratic with n² coefficient k/2, and constant third differences k give a cubic with n³ coefficient k/6. Geometric and Fibonacci-type sequences never reach constant differences. The second screen iterates uₙ₊₁ = f(uₙ), shows the behaviour on a cobweb diagram and adds terms in sigma notation. A few terms never prove a pattern; the sim needs two equal differences.
How to use
Choose a Pattern and press Add next pattern to grow it; the new pieces turn orange. Read the difference table, or untick Show differences and nth term and work them out first. Type coefficients into Your rule to test a formula, and enter a number in Is this a term? to check it. On Recurrence and sums, choose a Recurrence, move the sliders a, b, u₁ and N, and watch the cobweb diagram.
Parameters you can change
- Screen Patterns and the nth term, Recurrence relations and sums
- Pattern Matchstick squares, Matchstick triangles, Staircase of tiles, n × n matchstick grid, n × n × n cube, Sierpiński triangle, Fibonacci squares, Your own sequence
- Number of patterns (terms) 4–8 patterns
- Your own sequence (numbers separated by commas)
- Show the differences and the nth term
- Recurrence relation uₙ₊₁ = a·uₙ + b, uₙ₊₁ = 1 − 1/uₙ (periodic), uₙ₊₂ = uₙ₊₁ + uₙ (Fibonacci-type)
- Coefficient a in uₙ₊₁ = a·uₙ + b -2–2
- Constant b in uₙ₊₁ = a·uₙ + b -10–10
- First term u₁ -10–20
- Second term u₂ (Fibonacci-type) -10–20
- Number of terms added N 1–20 terms
Questions to explore
- How many matchsticks are needed for 50 squares in a row, and can exactly 100 matchsticks make a complete pattern?
- Why are the second differences of the staircase pattern constant, and what does that tell you about its nth term?
- For which values of a does uₙ₊₁ = a·uₙ + b converge, and what is the limit?
