Every linear-pattern question type — finding the rule Tₙ = dn + c, predicting a far term, working backwards to find which term equals a value, decreasing patterns, and patterns from diagrams — each worked in full, with a worksheet.
A number pattern with a constant difference between terms is called linear. Its rule always has the form Tn = dn + c, where d is the common difference and c is a constant. Find the rule once and you can jump to any term. This guide covers every way an exam uses it.
1The method to find the rule
- Find d — the constant difference between consecutive terms.
- Find c — substitute n = 1 into dn + c and match the first term.
- Write Tₙ = dn + c, then use it for any term.
2Type 1: Find the rule of an increasing pattern
- 1Common difference: 8 − 5 = 3, so d = 3.Each term goes up by 3.
- 2Tₙ = 3n + c. Test n = 1: 3(1) + c = 5, so c = 2.The rule must give the first term, 5.
- 3Rule: Tₙ = 3n + 2.Check n = 2: 3(2) + 2 = 8 ✓
3Type 2: Use the rule to find a far term
- 1Rule from Type 1: Tₙ = 3n + 2.Reuse the rule you found.
- 2T₅₀ = 3(50) + 2 = 150 + 2 = 152.Substitute n = 50 — no need to list 50 terms.
4Type 3: Work backwards: which term equals a value?
- 1Set the rule equal to 62: 3n + 2 = 62.You know the term value, want its position n.
- 23n = 60 → n = 20.Solve the linear equation.
- 3So 62 is the 20th term.Because n came out a whole number, 62 is in the pattern.
5Type 4: Decreasing patterns (negative d)
- 1Difference: 17 − 20 = −3, so d = −3.A decreasing pattern has a negative common difference.
- 2Tₙ = −3n + c. Test n = 1: −3 + c = 20, so c = 23.Match the first term.
- 3Rule: Tₙ = −3n + 23.Check n = 2: −6 + 23 = 17 ✓
6Type 5: Patterns from diagrams
- 1Count the difference: each new square adds 3 sticks, so d = 3.The constant difference comes from the repeating piece.
- 2Tₙ = 3n + c. Test n = 1: 3 + c = 4, so c = 1.The first shape uses 4 sticks.
- 3Rule: Tₙ = 3n + 1. So 10 squares need 3(10) + 1 = 31 sticks.Apply the rule to any number of squares.
The common difference is always the coefficient of n. If terms go up by 4, the rule starts with 4n; if they go down by 3, it starts with −3n.
c is not always the first term. Find it by substituting n = 1 into dn + c — don't just copy the first number. (For 5, 8, 11 the first term is 5 but c = 2.)
Linear patterns lead straight into arithmetic sequences in Grade 12, so this rule-finding skill keeps paying off. Practise every type below.
7Showing a pattern as a graph
A pattern can also be drawn on the Cartesian plane: plot the term number along the horizontal axis and the term value up the vertical axis. Each term becomes a point (n ; Tn).
A pattern with a constant difference always plots as points in a straight line — that is the visual signature of a linear rule Tn = dn + c.
Plot the points but do not join them with a solid line: term numbers are whole numbers only, so the data is discrete. A dotted guide line is fine to show the trend.
Practice exercises
Work each one out, then click to reveal the answer.
- 1Find the rule for 7, 10, 13, 16, … (find rule)
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d = 3; 3(1)+c = 7 → c = 4; Tₙ = 3n + 4 - 2Find the rule for 2, 6, 10, 14, … (find rule)
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d = 4; 4(1)+c = 2 → c = −2; Tₙ = 4n − 2 - 3For Tₙ = 5n + 1, find the 10th term. (far term)
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5(10)+1 = 51 - 4Find the 100th term of 4, 7, 10, … (far term)
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Tₙ = 3n+1; T₁₀₀ = 301 - 5Which term of 4, 7, 10, … equals 100? (work back)
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3n+1 = 100 → 3n = 99 → n = 33rd term - 6Which term of 5, 9, 13, … equals 81? (work back)
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4n+1 = 81 → n = 20th term - 7Find the rule for 30, 25, 20, 15, … (decreasing)
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d = −5; −5+c = 30 → c = 35; Tₙ = −5n + 35 - 8Find the rule for 12, 9, 6, 3, … (decreasing)
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d = −3; c = 15; Tₙ = −3n + 15 - 9The tile pattern below grows in the same way each time. Determine a rule for the number of tiles in the n-th term.
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d = 3, c = 2; Tₙ = 3n + 2 - 10Is 45 a term of 3, 7, 11, …? (check)
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4n−1 = 45 → 4n = 46 → n = 11,5, not whole → No
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