Page 23 - MAT KS3 Y8 Cambridge CheckPoint

2.2 Finding rules for sequences

2.2 Finding rules for sequences

This sequence of patterns is made from dots. Pattern 1 Pattern 2 Pattern 3
The numbers of dots used to make the
patterns form the sequence 5, 7, 9, …, …
As you go from one pattern to the next, two
more dots are added each time. The term-to-term 5 dots 7 dots 9 dots
rule is ‘add 2’.
You can use the term-to-term rule to work out Pattern 1 Pattern 2 Pattern 3
the position-to-term rule.
The term-to-term rule for this sequence is
‘add 2’, so start by listing the first three multiples
of 2 and compare them with the patterns of dots. 5 dots 7 dots 9 dots
The pattern is formed by adding multiples of 2,
Multiples of 2: 1 × 2 = 2 2 × 2 = 4 3 × 2 = 6
shown as red dots, to the three green dots at the
Number of dots: 2 + 3 = 5 4 + 3 = 7 6 + 3 = 9
start of each pattern.
The position-to-term rule for this sequence is:
term = 2 × position number + 3
Draw the next pattern, to check. Pattern 4
Pattern 4: term = 2 × 4 + 3 = 11 ✓ 11 dots

Worked example 2.2
For each sequence: i write down the term-to-term rule ii work out the position-to-term rule.
a 3, 6, 9, 12, …, … b 5, 9, 13, 17, …, …

a i Term-to-term rule is ‘add 3’. The numbers in the sequence are increasing by 3 each time.
ii Start by writing the sequence of numbers in a table.
Position number 1 2 3 4
The term-to-term rule is ‘add 3’, so write down the first four
Term 3 6 9 12 multiples of 3. The sequence matches exactly the multiples of 3.
Multiples of 3 3 6 9 12

Position-to-term rule is: Check that the rule works for the first few terms:
term = 3 × position number. 1st term = 3 × 1 = 3, 2nd term = 3 × 2 = 6, 3rd term = 3 × 3 = 9.

b i Term-to-term rule is ‘add 4’. The numbers in the sequence are increasing by 4 each time.
ii Start by writing the sequence of numbers in a table.
Position number 1 2 3 4
The term-to-term rule is ‘add 4’, so write down the first four
Term 5 9 13 17 multiples of 4.
Each term in the sequence is one more than a multiple of 4.
Multiples of 4 4 8 12 16
Position-to-term rule is: Check that your rule works for the first few terms:
term = 4 × position number + 1. 1st term = 4 × 1 + 1 = 5, 2nd term = 4 × 2 + 1 = 9,
3rd term = 4 × 3 + 1 = 13.

2 Sequences, expressions and formulae 21