Turning a pattern into a general rule — first described in words, then written algebraically as Tₙ = dn + c — and using that rule to find far-off terms or to work backwards, with worked examples, a worksheet with memo and a quiz.
Extending a pattern term by term is fine for the next two or three terms. But if an exam asks for the 100th term, you need a shortcut: a general rule that jumps straight to any term you like.
1From a table to a rule in words
Patterns are often given in a table, with the position number n on top and the term underneath. Start by describing in ordinary words what you do to the position number to get the term.
- 1The terms go up by 3 each time, so the rule involves multiplying n by 3.The constant difference tells you what n is multiplied by.
- 2Test 3 × n: for n = 1 that gives 3, but the term is 5 — it is 2 too small.Compare what you get with what you want.
- 3In words: 'multiply the position number by 3, then add 2'.That is a complete rule — check it on n = 4: 3(4) + 2 = 14 ✓.
2Writing the rule algebraically: Tₙ = dn + c
d = the constant difference • c = the number needed to fix the first term
(c is the 'zero-th term' — the term that would come before the first)
- 1Find d, the constant difference: d = 3.8 − 5 = 3.
- 2Find c: work backwards one step from the first term. 5 − 3 = 2, so c = 2.c is the term before the first one.
- 3Rule: Tₙ = 3n + 2.Substitute d = 3 and c = 2.
- 4Check with n = 3: 3(3) + 2 = 11 ✓Always test your rule on a term you were given.
The fastest way to find c: take the first term and subtract one difference. For 5, 8, 11 that is 5 − 3 = 2. It works every time for constant-difference patterns.
3Using the rule to find a far term
- 1The rule is Tₙ = 3n + 2.Found in the previous example.
- 2Substitute n = 50: T₅₀ = 3(50) + 2.'50th term' means n = 50.
- 3= 150 + 2 = 152.No need to write out 50 terms.
4Working backwards: which term equals a given value?
- 1Set the rule equal to 62: 3n + 2 = 62.The question gives the term and wants the position.
- 2Subtract 2 from both sides: 3n = 60.
- 3Divide both sides by 3: n = 20.
- 462 is the 20th term.Answer in words. If n had not been a whole number, 62 would not be in the pattern at all.
5Decreasing patterns have a negative d
- 1d = 17 − 20 = −3.The pattern decreases, so d is negative.
- 2c = first term − d = 20 − (−3) = 23.Subtracting a negative adds.
- 3Rule: Tₙ = −3n + 23.Check n = 2: −6 + 23 = 17 ✓
Tₙ = dn + c only works for constant-difference patterns. If the pattern has a constant ratio (like 2, 6, 18) or is made of square numbers, this formula does not apply — always check the differences first.
Practice exercises
Work each one out, then click to reveal the answer.
- 1Describe in words the rule for 4, 9, 14, 19, …
Show answer ▾Hide answer ▴
Multiply the position number by 5, then subtract 1 - 2Determine the general rule (Tₙ) for 4, 9, 14, 19, …
Show answer ▾Hide answer ▴
d = 5, c = 4 − 5 = −1 → Tₙ = 5n − 1 - 3Determine the general rule for 7, 12, 17, 22, …
Show answer ▾Hide answer ▴
d = 5, c = 2 → Tₙ = 5n + 2 - 4Determine the general rule for 6, 11, 16, 21, …
Show answer ▾Hide answer ▴
d = 5, c = 1 → Tₙ = 5n + 1 - 5Determine the general rule for 30, 26, 22, 18, …
Show answer ▾Hide answer ▴
d = −4, c = 34 → Tₙ = −4n + 34 - 6For Tₙ = 4n + 3, find the 10th term.
Show answer ▾Hide answer ▴
4(10) + 3 = 43 - 7For Tₙ = 6n − 2, find the 25th term.
Show answer ▾Hide answer ▴
6(25) − 2 = 148 - 8Which term of 5, 9, 13, … equals 45?
Show answer ▾Hide answer ▴
Tₙ = 4n + 1; 4n + 1 = 45 → n = 11th term - 9Is 100 a term of the pattern 3, 7, 11, …? Show your working.
Show answer ▾Hide answer ▴
Tₙ = 4n − 1; 4n − 1 = 100 → 4n = 101 → n = 25,25, not a whole number → No - 10A pattern of matchsticks is 4, 7, 10, … Determine a rule for the number of sticks in shape n.
Show answer ▾Hide answer ▴
d = 3, c = 1 → Tₙ = 3n + 1
Quick Quiz
4 quick questions on what you just read. Take it when you feel ready.
Now practise it
Download Grade 8 past papers and worksheets on this topic.