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2.1 Generating sequences

2.1 Generating sequences

In a linear sequence the terms increase (or decrease) by the same amount each time.
In a non-linear sequence the terms increase (or decrease) by a different amount each time.
Here is a non-linear sequence of numbers.
3, 6, 11, 18, 27, ...
Look at the di&erences between each of the terms.
3 + 3 = 6, 6 + 5 = 11, 11 + 7 = 18, 18 + 9 = 27, ...

Can you see that the di&erences between the terms are not the same?
!e term-to-term rule is ‘add 3, add 5, add 7, add 9, …’ .
!e position-to-term rule for this sequence is
‘term = position number + 2’. Position number 1 2 3 4 5
2
Check that this rule works by substituting the
position numbers given in the table. Term 3 6 11 18 27
For example: 2nd term = 2 + 2 = 4 + 2 = 6 9 4th term = 4 + 2 = 16 + 2 = 18 9
2
2
To "nd any other term in the sequence, substitute the position number into the rule.
For example: 6th term = 6 + 2 = 36 + 2 = 38
2

Worked example 2.1

a Are these sequences linear or non-linear? i 6, 4, 2, 0, −2, ... ii 8, 5, 1, −4, −10, ...
b The first term of a non-linear sequence is 4.
The term-to-term rule is multiply by 2.
Write down the first four terms of the sequence.
c The position-to-term rule of a non-linear sequence is: term = 2 × position number .
2
Work out the first four terms of the sequence.

a i The sequence is linear. The sequence is decreasing by the same amount (2) each time.
ii The sequence is non-linear. The sequence is decreasing by a different amount (3, 4, 5, 6, ...)
each time.
b First four terms are 4, 8, 16, 32. Write down the first term, which is 4, then use the
term-to-term rule to work out the next three terms.
Second term = 4 × 2 = 8, third term = 8 × 2 = 16,
fourth term = 16 × 2 = 32.
c First four terms are 2, 8, 18, 32. Use the position-to-term rule to work out each term.
First term = 2 × 1 = 2 × 1 = 2, second term = 2 × 2 = 2 × 4 = 8,
2
2
2
2
third term = 2 × 3 = 2 × 9 = 18, fourth term = 2 × 4 = 2 × 16 = 32.
) Exercise 2.1

1 Write down whether each sequence is linear or non-linear.
Explain your answers.
a 11, 15, 19, 23, 27, ... b 20, 30, 40, 50, 60, ... c 4, 5, 7, 10, 14, ...
d 20, 18, 15, 11, 6, ... e 100, 95, 90, 85, 80, ... f 10, 7, 1, −8, −20, ...
1
1
1
1
1
g 0.5, 1, 1.5, 2, 2.5, ... h 3 , 5 , 9 , 17 , 33 , ... i 20, 12, 4, −4, −12, ...
2 2 2 2 2
16 2 Sequences and functions