Grade 9 Number Patterns: Short Notes

9 ශ්‍රේණිය · 3 පරිච්ඡේදය · සංඛ්‍යා රටා

93%

of Grade 9 term tests include it (37 of 40)

309

marks in past papers (68 questions)

10 min

full lesson with quick checks

🎯 What you'll learn

Find the common difference of a number pattern
Write the general term Tₙ of a pattern that goes up or down by a fixed amount
Find any term from the general term
Find which term has a given value, or show that a number is not a term
Use general terms for shape patterns and special patterns (odd, square, triangular numbers)

Chairs for prize-giving

For the prize-giving in your school hall, row 1 has 12 chairs and every row behind has 3 more than the row in front. The principal asks: which row will have 57 chairs? You could draw all the rows… or find one rule that gives any row instantly.

Terms and the common difference

In the pattern 5, 8, 11, 14, … each number is a term.T₁ = 5 is the 1st term, T₂ = 8 the 2nd, and Tₙ the nth term.The terms go up by 3 each time: the common difference is 3.Difference = any term − the term before it. If the terms go down, the difference is negative.

Where does Tₙ = 3n + 2 come from?

The difference is 3, so compare with the 3 times table: 3, 6, 9, 12. Every term is 2 more. So Tₙ = 3n + 2.

🧮 The general term

d is the common difference and T₁ the first term. The number in front of n is always the difference; the extra number is the first term minus the difference. Always test it with n = 1.

Tₙ = d × n + (T₁ − d)

✏️ Find Tₙ, then the 20th term

For the pattern 7, 11, 15, 19, … find (i) the general term (ii) the 20th term.

1Difference d = 11 − 7 = 4.
2Extra number = T₁ − d = 7 − 4 = 3, so Tₙ = 4n + 3.
3Test: n = 2 gives 8 + 3 = 11 ✓
4(ii) T₂₀ = 4 × 20 + 3 = 83.

✅ (i) Tₙ = 4n + 3 (ii) 83

⚠️ Using the first term as the extra number

For 5, 8, 11, … many students write Tₙ = 3n + 5. Test n = 1: 3 + 5 = 8, but the first term is 5. ✗ The extra number is T₁ − d = 5 − 3 = 2, giving 3n + 2. Substituting n = 1 takes five seconds and catches this every time.

✏️ A pattern that goes down

Find the general term of 30, 26, 22, 18, … and its 9th term.

1d = 26 − 30 = −4 (negative, because the terms go down).
2Extra number = 30 − (−4) = 34.
3Tₙ = −4n + 34, usually written 34 − 4n. Test n = 1: 30 ✓
4T₉ = 34 − 36 = −2.

✅ Tₙ = 34 − 4n, T₉ = −2

⚠️ n starts at 1, and must be whole

Two slips from term tests: finding the first term with n = 0 (for 25 − 3n that gives 25, but T₁ = 22), and accepting n = 17⅓ as "the 17th term". There is no first term at n = 0 and no 17⅓th term. If n is not a whole number, the value is not in the pattern.

A matchstick pattern

Each new square needs 3 more sticks, so the stick counts 4, 7, 10, … have difference 3 and Tₙ = 3n + 1.

Special patterns to know

• Even numbers 2, 4, 6, … Tₙ = 2n• Odd numbers 1, 3, 5, … Tₙ = 2n − 1• Square numbers 1, 4, 9, 16, … Tₙ = n²• Triangular numbers 1, 3, 6, 10, 15, … (add 2, then 3, then 4 …) Tₙ = n(n + 1)/2• Fractions 1/2, 2/3, 3/4, … Tₙ = n/(n + 1): look at top and bottom separately.

📌 Remember

• Common difference d = any term − the term before.• Tₙ = d × n + (T₁ − d). Test with n = 1.• To find a term, substitute n. To find which term, solve Tₙ = value.• If n is not a whole number, the value is not a term.• Know 2n, 2n − 1, n² and n(n + 1)/2.

🧾 Quick revision sheet

Common difference	Any term minus the term before it
General term from d and T₁	Tₙ = d × n + (T₁ − d)
General term of odd numbers	2n − 1
General term of square numbers	n²
First five triangular numbers	1, 3, 6, 10, 15 (Tₙ = n(n + 1)/2)
How do you show 42 is not a term of 3n − 10?	3n − 10 = 42 gives n = 17⅓, not a whole number.
Tₙ of 30, 26, 22, …	34 − 4n