3. Sequences
Definitions:
09/06/2023
Sequence A sequence is a list of numbers in a special order.
Term Each number in a sequence is called a term.
Term to Term Rule Defines the value of each term in a sequence
Linear Sequence
A sequence that increases/decreases by the same
amount each time.
4. Sequences 1
Discussion 1:
Sam is a biologist for a seed. She records the growth of Brand X bean
plants over six days.
Sam notices that Brand X bean plants grow to a number pattern.
She decides to create a table to show this.
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Day 1 Day 2 Day 3 Day 4 Day 5
5. plants over six days.
Sam notices that Brand X bean plants grow to a number pattern.
She decides to create a table to show this.
The number pattern can be written as a sequence.
Day 1 Day 2 Day 3 Day 4 Day 5
Day 1 2 3 4 5
Number of leaves 2 4 6 8 10
6. Sequences 1
Discussion 2:
Look at this Spanish tile pattern and determine the sequence that was
used and find the next five terms.
The sequence for the pattern is 1, 5, 9, 13…
The term-to-term rule:
The next five terms are
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1 tile 5 tiles 9 tiles 13 tiles
17 21 25 29 33
Starts at 1 and add 4
7. Sequences 1
Discussion 3:
a) Write a sequence for the number of dots in each term as the
sequence progresses,
b) What is the term-to-term rule for the sequence?
c) Use the rule to extend the sequence to the 5th term
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1, 3, 5, ........
Starts at 1, and add 2
1 3 5 7 9
8. Sequences
Examples:
09/06/2023
eg1 Find the T2T rule and the next
two terms of the sequence.
3, 5, 7, 9 …
eg2 Find the T2T rule and the next
two terms of the sequence.
2, 8, 32, 128 …
Try Find the T2T rule and the next two
terms of the sequence.
25, 28, 31, 34 …
Try Find the T2T rule and the next two
terms of the sequence.
2, 6, 18, 54 …
T2T Rule: +2
Next 2 terms: 11, 13 …
T2T Rule: +3
Next 2 terms: 37, 40 …
T2T Rule: × 4
Next 2 terms: 512, 2048 …
T2T Rule: +3
Next 2 terms: 162, 486 …
9. Task 1:
1 Find the term to term rule and the next two terms in the sequence.
a 4, 6, 8, 10 … b 5, 7, 9, 11 … c 2, 9, 1, 1, 7 …
There are non examples
d 2, 4, 8, 16 … e 7, 10, 13, 16 … f 21, 19, 17, 15 …
g 1024, 256, 64, 16 … h 0, 6, 12, 18 … i −9, −3, 3, 9 …
j 1, 4, 9, 16, 25 … k −3, 3, 13, 27 … l 0, 6, 16, 30 …
m 2𝑥, 4𝑥, 6𝑥, 8𝑥 … n 2𝑥, 4𝑥, 8𝑥, 16𝑥 … o 2𝑥 + 2, 3𝑥 + 5, 4𝑥 + 8
+𝟐
12, 14 …
+𝟐
13, 15 …
𝑵𝒐𝒏 − 𝑬𝒙𝒂𝒎𝒑𝒍𝒆
× 𝟐
32, 64 …
+𝟑
19, 22 …
−𝟐
13, 11 …
÷ 𝟒
4, 1 …
+𝟔
24, 30 …
+𝟐
15, 21 …
+𝟑, +𝟓, +𝟕 …
36, 49 …
+𝟔, +𝟏𝟎, +𝟏𝟒 …
45, 67 …
+𝟔, +𝟏𝟎, +𝟏𝟒 …
48, 70 …
+𝟐𝒙
10𝑥, 12𝑥 …
× 𝟐
32𝑥, 64𝑥 …
+(𝒙 + 𝟑)
5𝑥 + 11, 6𝑥 + 14 …
2 Find the term to term rule and the draw the next picture.
+3 𝑠𝑞𝑢𝑎𝑟𝑒𝑠
3 Is 318 a term in the sequence 1, 9, 17, 25…?
No the terms are all odd numbers.
10. Sequences
Examples:
09/06/2023
eg3 Use the T2T rule to find the
first 5 terms in the sequence.
Starts at 6, +𝟑
Try Use the T2T rule to find the
first 5 terms in the sequence.
Starts at 6, −𝟑
6 9 12 15 18 6 3 0 -3 -6
11. Task 2:
1 Use the T2T rule to find the first 5 terms in the sequence.
a 𝑆𝑡𝑎𝑟𝑡: 𝟒 𝑅𝑢𝑙𝑒: +𝟑 b 𝑆𝑡𝑎𝑟𝑡: 𝟓 𝑅𝑢𝑙𝑒: +𝟑 c 𝑆𝑡𝑎𝑟𝑡: 𝟓 𝑅𝑢𝑙𝑒: +𝟒
4, 7, 10, 13, 16 5, 8, 11, 14, 17 5, 9, 13, 17, 21
d 𝑆𝑡𝑎𝑟𝑡: 𝟏 𝑅𝑢𝑙𝑒: × 𝟑 e 𝑆𝑡𝑎𝑟𝑡: 𝟓 𝑅𝑢𝑙𝑒: −𝟒 f 𝑆𝑡𝑎𝑟𝑡: 𝟏𝟎 𝑅𝑢𝑙𝑒: −𝟓
1, 3, 9, 27, 81 5, 1, −3, −7, −11 10, 5, 0, −5, −10
g 𝑆𝑡𝑎𝑟𝑡: 𝟏 𝑅𝑢𝑙𝑒: × 𝟐 𝑎𝑛𝑑 + 𝟏
1, 3, 7, 15, 31, 63
h 𝑆𝑡𝑎𝑟𝑡: 𝟑𝟑 𝑅𝑢𝑙𝑒: +𝟏 𝑎𝑛𝑑 ÷ 𝟐
33, 17, 9, 5, 3
i 𝑆𝑡𝑎𝑟𝑡: 𝒙 + 𝟓 𝑅𝑢𝑙𝑒: +(𝟐𝒙 + 𝟑)
𝑥 + 5, 3𝑥 + 8, 5𝑥 + 11, 7𝑥 + 14, 9𝑥 + 17
j 𝑆𝑡𝑎𝑟𝑡: 𝟐𝒙 𝑅𝑢𝑙𝑒: × (𝟐𝒙)
2𝑥, 4𝑥2
, 8𝑥3
, 16𝑥4
, 32𝑥5
2
The rule is to add four.
Explain why she has not described
the rule for a sequence.
The first term of her sequence is a
factor of 18. Come up with 3
different sequences.
Sofia is writing a sequence. a
b
16. Sequences 2
Discussion:
09/06/2023
Take a look at the sequence
7, 11, 15, 19 …
Find the term to term rule. +4
Find the next two terms. 23 27
Find the 10th term. 31 35 39 43
Find the 80th term.
It will take a long time to find this
term using the term to term rule.
We can use the position to term rule.
4𝑛 + 3
= 4 80 + 3
= 320 + 3
= 323
17. Sequences 2
Definitions:
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Position to Term Rule a rule that defines the value of each term in a
sequence with regard to its position.
nth Term Rule Also known as the position to term rule.
Uses algebra to work out the term given the position.
𝑛 The position of the nth term.
Position 1 2 3 4 … 𝑛
Sequence 7 11 15 19 … 𝟒𝒏 + 𝟑
nth term rule
18. Sequences
Examples:
09/06/2023
eg1 Use the nth term rule to find
the first four terms.
𝟐𝒏
eg2 Use the nth term rule to find
the first four terms.
𝟐𝒏 + 𝟒
Try Use the nth term rule to find
the first four terms.
𝟓𝒏
Try Use the nth term rule to find
the first four terms.
𝟓𝒏 − 𝟑
𝑛 1 2 3 4
2𝑛 2 4 6 8
𝑛 1 2 3 4
5𝑛 5 10 15 20
𝑛 1 2 3 4
2𝑛 + 4 6 8 10 14
𝑛 1 2 3 4
5𝑛 − 3 2 7 12 17
19. Task 2:
1 Find the first 4 terms of each sequence.
a 3𝑛 b 4𝑛 c 4𝑛 + 1
d 4𝑛 + 7 e 4𝑛 − 10 f 𝑛 − 10
3, 6, 9, 12 4, 8, 12, 16 5, 9, 13, 17
11, 15, 19, 23 −6, −2, 2, 6 −9, −8, −7, −6
g 𝑛 + 10 h 1
2
𝑛 + 10 i 1
2
𝑛 − 2
11, 12, 13, 14 10.5, 11, 11.5, 12 −1.5, −1, −0.5, 0
2 Match the rule to the sequence.
9, 12, 15, 18 7, 8, 9, 10 6, 12, 18, 24 9, 15, 21, 27
6𝑛 𝑛 + 6 6𝑛 + 3 3𝑛 + 6
3 Find the first 4 terms of each sequence.
a −2𝑛 b 3 − 2𝑛 c 3 − 4𝑛
−2, −4, −6, −8 1, −1, −3, −5 −1, −5, −9, −13
d 𝑛2 e 𝑛3 f 3𝑛
1, 4, 9, 16 1, 8, 27, 64 3, 9, 27, 81
23. Sequences 3
Starter:
09/06/2023
Sequences
Which sequences below are linear?
1) 2, 5, 8, 11 …
2) 24, 19, 14, 9 …
3) 3, 6, 12, 24 …
4) 120𝑥, 60𝑥, 30𝑥, 15𝑥 …
Write the term to term rule each
sequence.
1) 4, 14, 24, 34 …
2) 1, 4, 16, 64 …
Use the position to term rule to write
the first four terms of the sequence.
𝟓𝒏 − 𝟐
Use the nth term rule to find the 8th
term of the sequence.
𝟔𝟎 − 𝟓𝒏
24. Sequences 3
Starter:
09/06/2023
Sequences
Which sequences below are linear?
1) 2, 5, 8, 11 … Yes
2) 24, 19, 14, 9 … Yes
3) 3, 6, 12, 24 … No
4) 120𝑥, 60𝑥, 30𝑥, 15𝑥 … No
Write the term to term rule each
sequence.
1) 4, 14, 24, 36 … + 10
2) 1, 4, 16, 64 … × 4
Use the position to term rule to write
the first four terms of the sequence.
𝟓𝒏 − 𝟐
Use the nth term rule to find the 8th
term of the sequence.
𝟔𝟎 − 𝟓𝒏
3, 8, 13, 18 … 20
25. Sequences 3
Discussion:
09/06/2023
What is the nth term rule for this sequence?
6 12 18 24 …
+6 +6 +6 +6
Position (𝒏) 1 2 3 4 … 𝑛
Sequence 6 12 18 24 … 6𝑛
What is the nth term rule for this sequence?
4 10 16 22 …
+6 +6 +6 +6
Position (𝒏) 1 2 3 4 … 𝑛
Sequence 4 10 16 20 … 6𝑛 − 2
