Number Patterns: Linear Sequences (Grade 10)

Finding the constant difference, writing the general term Tₙ = dn + c, calculating any term, and working backwards to find which term has a given value.

A sequence is linear (or arithmetic) when you add the same amount each time. That amount is the constant difference, d.

1The general term

Tₙ = dn + c

d = constant difference
c = the term BEFORE the first one (that is, T₀)

The sequence 5 ; 8 ; 11 ; 14 ; 17 plotted against n — linear sequences lie on a straight line.

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Because Tₙ = dn + c is the equation of a straight line, the constant difference d is exactly the gradient. That is why the plotted points are collinear.

Worked ExampleDetermine the general term of 5 ; 8 ; 11 ; 14 ; …

1
d = 8 − 5 = 3. Check: 11 − 8 = 3 and 14 − 11 = 3. ✓
Always check the difference is the same all the way.
2
Tₙ = 3n + c.
Substitute d = 3.
3
T₁ = 5, so 3(1) + c = 5.
Use the first term to find c.
4
c = 2, so Tₙ = 3n + 2.
Check T₃: 3(3) + 2 = 11. ✓

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c is not the first term. For 5 ; 8 ; 11 the first term is 5 but c = 2 — it is the value the pattern would have had at n = 0.

2Finding a specific term

Worked ExampleCalculate T₅₀ for the sequence 5 ; 8 ; 11 ; …

1
Tₙ = 3n + 2.
2
T₅₀ = 3(50) + 2.
Substitute n = 50.
3
= 152.

3Working backwards

Worked ExampleWhich term of 5 ; 8 ; 11 ; … is equal to 47?

1
3n + 2 = 47.
Set the general term equal to the value.
2
3n = 45.
3
n = 15, so 47 is the 15th term.
Check: 3(15) + 2 = 47. ✓

Worked ExampleIs 100 a term of the sequence 5 ; 8 ; 11 ; …?

1
3n + 2 = 100 → 3n = 98.
2
n = 983 = 32,67…
3
n is not a whole number, so 100 is not a term of this sequence.
Term numbers must be positive integers — say this in words.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Write down the constant difference of 5 ; 8 ; 11 ; 14 ; …
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d = 3
2
Determine the general term Tₙ of 5 ; 8 ; 11 ; 14 ; …
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d = 3, T₁ = 5 → c = 2 → Tₙ = 3n + 2
3
Calculate T₅₀ for the sequence 5 ; 8 ; 11 ; …
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3(50) + 2 = 152
4
Which term of 5 ; 8 ; 11 ; … is equal to 47?
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3n + 2 = 47 → n = 15
5
Is 100 a term of the sequence 5 ; 8 ; 11 ; …? Justify.
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3n = 98 gives n = 32,67 which is not a whole number, so no.
6
Determine the general term of 7 ; 11 ; 15 ; 19 ; …
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d = 4, c = 3 → Tₙ = 4n + 3
7
Determine the general term of 20 ; 17 ; 14 ; 11 ; …
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d = −3, c = 23 → Tₙ = −3n + 23
8
Explain why a linear sequence plots as a straight line.
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Because Tₙ = dn + c is the equation of a straight line, with the constant difference as the gradient.