Mathematics notes, chapter 1

Patterns and Sequences

Textbook Mathematics Form 2 (DLP), pages 1–17

What this chapter covers

Recognise and describe patterns of various sets of numbers and objects in everyday life (1.1.1)
Explain the meaning of a sequence (1.2.1)
Identify and describe the pattern of a sequence, then complete and extend the sequence (1.2.2)
Describe the pattern of a sequence using numbers, words and algebraic expressions (1.3.1)
Determine specific terms of a sequence (1.3.2)
Solve problems involving sequences (1.3.3)

1.1Patterns

Patterns are lists of numbers or objects arranged according to a rule or design. To find the rule, compare each term with the one before it.

Patterns of objects

For objects, look at what is added (or taken away) from one picture to the next. Example 1 (textbook p. 3): add two dots to the previous object, or add one triangle to the previous object.

Four patterns of squares with 1, 4, 7 and 10 squares: a middle square with an arm of squares to the left, to the right and upwards — Each pattern adds one square to each of three arms: 1, 4, 7, 10, … The rule is add 3 (textbook p. 3, grid-paper activity).

Patterns of numbers

A number pattern is made by adding, subtracting, multiplying or dividing the previous number.

Numbers	Rule	Operation
−10, −4, 2, 8, …	Add 6	+6
17, 7, −3, −13, …	Subtract 10	−10
2, 6, 18, 54, …	Multiply by 3	×3
81, 27, 9, 3, …	Divide by 3	÷3
1, 1½, 2, 2½, …	Add ½	+½
−2.3, −2.6, −2.9, −3.2, …	Subtract 0.3	−0.3

Example 2 (textbook p. 4): the six sequences in the table.

With negative numbers, check the direction. From −10 to −4 the number goes up by 6, so the rule is add 6, not subtract 6.

Odd and even numbers in a pattern

Even numbers can be divided by 2 exactly (2, 4, 6, 8, …). Odd numbers cannot (1, 3, 5, 7, 9, …).

Example 3 (textbook p. 4): 7, 12, 17, 22, 27, …, 67

Odd numbers: 7, 17, 27, 37, 47, 57, 67 (add 10)
Even numbers: 12, 22, 32, 42, 52, 62 (add 10)
The terms go up by 5 each time, so odd and even terms take turns. Taking every second term gives a rule of add 10.

Pascal's Triangle

Every row starts and ends with 1. Every other number is the sum of the two numbers above it.

Textbook picture of Pascal's Triangle with plus signs and arrows showing how each number is the sum of the two above — How each number is made from the two above it.

Picture: Mathematics Form 2 (DLP) textbook, p. 5.

Method 1: the diagonal 1, 2, 3, 4, … adds 1 each time.
Method 2: the diagonal 1, 3, 6, 10, … adds 2, 3, 4, … (the triangular numbers).
Row 4 of the triangle is 1, 3, 3, 1; row 5 is 1, 4, 6, 4, 1; row 6 is 1, 5, 10, 10, 5, 1; row 7 is 1, 6, 15, 20, 15, 6, 1 (Example 4).

Chinese mathematicians knew this pattern as Yang Hui's Triangle.

Fibonacci Numbers

Fibonacci Numbers begin 0, 1, 1 and each next term is the sum of the two terms before it: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …

Boxes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 with arrows from each pair to their sum — 0 + 1 = 1, 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, … (textbook p. 6).

Example 5 (textbook p. 6): complete 1, 3, __, __, 11, …

Add the two before: 1 + 3 = 4, 3 + 4 = 7, 4 + 7 = 11. The sequence is 1, 3, 4, 7, 11, …

Real life: the seeds of a sunflower form spirals of 21 and 34. Both are Fibonacci Numbers.

Complete the Fibonacci-type sequence 2, 5, __, 12, __, …

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2 + 5 = 7, then 5 + 7 = 12 (checks), then 7 + 12 = 19. The sequence is 2, 5, 7, 12, 19, …

1.2Sequences

A sequence is a set of numbers or objects arranged according to a certain pattern. The numbers are written in order and every term follows the same rule.

Example 6 (textbook p. 8): is it a sequence?

(a) −10, −6, −2, 2, 6, … : add 4 every time, so it is a sequence.
(b) 4, 5, −7, 10, −14, … : the changes are +1, −12, +17, −24, so there is no single pattern. It is not a sequence.

Completing a sequence

Find the rule from two terms that are next to each other, then check it on the others. Work forwards by applying the rule, and backwards by doing the opposite operation.

Sequence	Rule	Missing terms
7, 13, __, 25, __, …	+6	19, 31
88, __, 64, 52, __, __, …	−12	76, 40, 28
__, 0.3, __, 0.027, 0.0081, __, …	×0.3	1, 0.09, 0.00243
__, __, ⅓, ⅔, __, …	+⅓	−⅓, 0, 1

Example 7 (textbook p. 8). To find a missing term before the first number you are given, do the opposite operation: 0.3 ÷ 0.3 = 1, and ⅓ − ⅓ = 0.

Example 8 (textbook p. 9): start with 96 and subtract 4 each time

96, 92, 88, 84, 80, … Multiply by 3 starting from 7: 7, 21, 63, 189, 567, … Divide by 5 starting from 400: 400, 80, 16, 3.2, 0.64, …

Dividing by 5 is not the same as subtracting 5. Test the rule on the second and third terms too (400, 80, 16): 400 ÷ 5 = 80 and 80 ÷ 5 = 16.

Complete 13.3, 5.3, −2.7, __, −18.7, … and state the rule.

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The rule is subtract 8: −2.7 − 8 = −10.7, then −10.7 − 8 = −18.7. ✓

1.3Patterns and sequences

Numbers, words and algebraic expressions

Example 9 (textbook p. 10): the sequence 1, 9, 17, 25, 33, … can be described in three ways.

Way	Description
Numbers	+8 each time
Words	Add 8 to the previous number
Algebraic expression	1 + 8n, where n = 0, 1, 2, 3, …

An algebraic expression uses letters for numbers that change. In a sequence, n is the position and the expression gives the term at that position.

Write 3, 6, 9, 12, 15, … as an algebraic expression

Add 3 each time, so the terms are multiples of 3: 3n, with n = 1, 2, 3, …
Check: n = 1 gives 3, n = 4 gives 12. ✓
If a sequence starts at another number, adjust: 5, 8, 11, 14, … is 3n + 2 (n = 1 gives 5, n = 4 gives 14).

Multiplying patterns use powers: 2, 4, 8, 16, … is 2ⁿ with n = 1, 2, 3, …, and 100, 50, 25, 12.5, … is 100 ÷ 2ⁿ with n = 0, 1, 2, 3, ….

Always say which numbers n takes. n = 0, 1, 2, … and n = 1, 2, 3, … give different expressions for the same sequence (1 + 8n and 8n − 7 both give 1, 9, 17, …).

Terms of a sequence

The nth term is written T_n. T is the term and n is its position. For 4, 8, 12, 16, …: T₁ = 4, T₂ = 8, T₃ = 12, T₄ = 16.

Example 10 (textbook p. 11): the 5th term of 2, 10, 18, …

Rule: add 8. T₁ = 2, T₂ = 10, T₃ = 18, T₄ = 26, T₅ = 34.

Example 11 (textbook p. 11): which term of 65, 60, 55, 50, … is 40?

Rule: subtract 5. T₅ = 45, T₆ = 40. So 40 is the 6th term.

A quicker check for a sequence that adds d each time: T_n = first term + (n − 1) × d. For 65, 60, …: 65 − (n − 1) × 5 = 40 gives n − 1 = 5, so n = 6.

Counting from T₁, you add the rule one time fewer than the position. T₅ needs 4 additions, not 5.

For −3, 5, 13, … find T₇.

Show answerHide answer

Rule: add 8. T₇ = −3 + 6 × 8 = 45.

Solving problems

Step	Example 12 (textbook p. 12): fish feeder
Understand	Fish are fed 4 times a day, first at 7:35 a.m. When is the third feeding?
Plan	1 day = 24 hours, so the gap is 24 ÷ 4 = 6 hours. The feeding times form a sequence that adds 6 hours.
Do	T₁ = 7:35 a.m., T₂ = 1:35 p.m., T₃ = 7:35 p.m.
Conclude	The third feeding is at 7:35 p.m.

Buses leave every 30 minutes. Bus A leaves at 8:00 a.m. At what time does Bus E leave?

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A 8:00, B 8:30, C 9:00, D 9:30, E 10:00 a.m. (four gaps of 30 minutes = 2 hours).

Summary

Idea	Remember
Pattern	A rule for objects or numbers: +, −, ×, ÷ the previous term
Pascal's Triangle	Each number is the sum of the two above; rows start and end with 1
Fibonacci Numbers	0, 1, 1, 2, 3, 5, 8, … each term is the sum of the previous two
Sequence	Numbers or objects that follow one pattern
Describe a pattern	Numbers (+3), words (add 3), algebraic expression (3n)
Term	T_n is the term at position n
Problems	Understand, plan, do, conclude; check by listing terms

Content and examples follow Mathematics Form 2 (DLP) textbook, Chapter 1, pp. 1–17.