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Number Sequences

Number Sequences

Sequences from patterns and tables

With Table Pattern Core + Extended 7 min read

On this page
  1. In this note
  2. Formula bank
  3. From picture to table to formula
  4. Patterns with two kinds of object
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 2.7: sequences from patterns and tables
  • 2.7Continue a pattern of shapes and complete a table of values.
  • 2.7Find an expression for the number of objects in Pattern n, and explain it from the picture.
  • 2.7Answer: which pattern uses a given number, the largest pattern you can make, comparing two parts of a pattern.

Before you start: linear sequences and special sequences.

Formula bank

Pattern → formulaLearn itcount Patterns 1, 2, 3 → table → nth termUse it when a sequence of diagrams.
Which pattern uses X?Learn itsolve nth term \(=X\)n must be a whole number, otherwise no pattern uses X.Use it when "which pattern has 60 tiles".
Largest pattern you can makeLearn itsolve nth term \(\le X\), round n DOWNUse it when "she has 150 dots".
Square with a ringLearn itinside \(n^{2}\), whole \((n+2)^{2}\), ring \(4n+4\)Use it when a square of tiles with a border.

From picture to table to formula

Each pattern in a sequence of diagrams is a term. Count the objects in Patterns 1, 2, 3, put the counts in a table, then find the nth term from the numbers.

Matchstick patternsPattern 1Pattern 2Pattern 3
Matchstick squares in a row: 4, 7, 10 sticks.
Pattern n1234n
Matchsticks471013\(3n+1\)

General methodHow to answer any pattern question

  1. Count Patterns 1, 2, 3 (draw the next one if asked). Put the counts in a table.
  2. Find the nth term from the numbers (linear: \(dn+c\); otherwise compare with \(n^{2}\)).
  3. Explain it from the picture: what does each new pattern add? (one stick to start, then 3 per square → 1 + 3n.)
  4. Use it: substitute n, or solve = (which pattern?) or ≤ (largest pattern, round down).

Example Pattern n uses 3n + 1 matchsticks. Sam has 100. What is the largest pattern he can make?

Solve ≤ 100 and round down.

  1. \(3n+1\le100\Rightarrow n\le33\)

Answer Pattern 33 (uses exactly 100)

Try it yourself

1
Pattern n has 5n + 1 dots. Dots in Pattern 3?
Show answer

Answer 16

2
Pattern n uses 6n + 2 sticks. Which pattern uses 200?
Show answer

Answer Pattern 33

3
Pattern n uses 3n + 1 sticks. Ali has 75. Largest pattern?
Hint
Round down: Pattern 25 needs 76.
Show answer

Answer Pattern 24

Patterns with two kinds of object

Tile patternsPattern 1Pattern 2Pattern 3
Blue tiles in the middle, white tiles round the edge.
Pattern n12345n
blue tiles1491625\(n^2\)
white tiles812162024\(4n+4\)
total916253649\((n+2)^2\)

The picture explains each formula: the blue block is \(n\) by \(n\); the whole pattern is \((n+2)\) by \((n+2)\); so the white ring is the difference \((n+2)^2-n^2=4n+4\).

Example A pattern uses 225 tiles in total. Which pattern is it?

Total \(=(n+2)^{2}\).

  1. \(n+2=15\)

Answer Pattern 13

Ways they test it

Type 1 Draw, table, formula

"Draw Pattern 4; complete the table; find an expression for Pattern n." (4 marks)

Spot it a sequence of diagrams.

  1. Count and tabulate (B1, B1).
  2. nth term (B2).

For example 4, 7, 10, 13 sticks

  1. d = 3, zero term 1

Answer \(3n+1\)

Type 2 Largest pattern

"Nada has 150 dots. Find the largest pattern she can make." (2 marks)

Spot it a limited number of objects.

  1. Solve ≤.
  2. Round DOWN to a whole pattern.

For example \(4n+1\le150\)

  1. \(n\le37.25\)

Answer Pattern 37

Type 3 Explain

"Explain why Pattern n has 4n + 1 dots." / "Explain why no pattern uses 100 sticks."

Spot it the word explain.

  1. Refer to the picture (what each pattern adds).
  2. Or: solving gives a non-whole n.

Type 4 Two quantities

"Find the first pattern with more black beads than twice the white beads."

Spot it two counts, one often quadratic.

  1. Extend the table until the condition is true.

Mistakes to avoid

  • ✗ 37.25 patterns✓ Pattern 37Whole patterns only - round down when you are limited.
  • ✗ 'It goes up by 4' as the expression✓ \(4n+1\)An expression in n is needed.
  • ✗ Formula built from Patterns 2, 3, 4 but called Pattern 1✓ check n = 1Positions must match.

Summary

  • Picture → table → nth term → answer.
  • Explain the formula from what each pattern adds.
  • 'Which pattern uses X': solve =, n must be whole.
  • 'Largest you can make': solve ≤, round down.

If you see …

  • "has 150 … largest"inequality, round down.
  • "explain why no pattern"solve and show n is not whole.

Practice

Practice: easy, medium, hard

Easy

1
Counts 8, 14, 20, 26. Expression for Pattern n?
Show working
  1. d = 6, zero term 2

Answer \(6n+2\)

2
Pattern n has 3n + 1 sticks. How many in Pattern 20?
Show working
  1. \(3(20)+1\)

Answer 61

3
Pattern 1 has 5 dots, Pattern 2 has 9, Pattern 3 has 13. Dots in Pattern 4?
Show answer

Answer 17

Medium

1
White tiles 4n + 4. Which pattern has 60 white tiles?
Guide me

What is it asking? n

What do we know? \(4n+4=60\)

Plan solve

Show working
  1. \(4n=56\)

Answer 14

2
Pattern n uses 4n + 1 dots. Nada has 150 dots. Largest pattern?
Guide me

What is it asking? n

What do we know? \(4n+1\le150\)

Plan round down

Show working
  1. \(n\le37.25\)

Answer 37

3
Ali has 75 sticks; Pattern n uses 3n + 1. How many sticks are left after the largest pattern?
Guide me

What is it asking? sticks left

What do we know? largest pattern first

Plan n = 24 uses 73

Show working
  1. 75 − 73

Answer 2

Hard

1
Pattern n uses 6n + 2 sticks. Explain why no pattern uses 100 sticks.
Guide me

What is it asking? a reason

What do we know? solve \(6n+2=100\)

Plan is n whole?

Show working
  1. \(6n=98\)
  2. \(n=16.3\ldots\)

Answer \(6n+2=100\) gives \(n=16.33\ldots\), not a whole number

PartAnswer / workingMarksGuidance
solves the equationM1
n not wholeA1
Total: 2 marks
2
A tile pattern uses 225 tiles in total, where Pattern n uses (n + 2)² tiles. Which pattern?
Guide me

What is it asking? n

What do we know? \((n+2)^{2}=225\)

Plan square root

Show working
  1. \(n+2=15\)

Answer 13

PartAnswer / workingMarksGuidance
\(\sqrt{225}=15\)M1
13A1
Total: 2 marks
3
White beads \(2n+3\), black beads \(\tfrac{(n+1)(n+2)}2\). First pattern with more than twice as many black as white?
Guide me

What is it asking? n

What do we know? compare black with 2 × white

Plan extend the table

Show working
  1. n = 6: 28 < 30
  2. n = 7: 36 > 34

Answer 7

PartAnswer / workingMarksGuidance
trial of n = 6 and 7M1
7A1
Total: 2 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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