Mastering Sequences in Mathematics

 
Year 8
Lockdown lessons
 
Week 6: Lesson 1
 
Introduction to sequences
What is a sequence?
 
A mathematical sequence is a series of numbers connected by a rule.
This rule defines their pattern
 
3, 8, 13, 18, 23
 
+5 every time
 
2, 4, 8, 16, 32
 
x2 every time
 
11, 7, 3, -1, -5
 
-4 every time
 
Each part of a sequence is
called a ‘term’
 
How we move from one term to the next is
called the 
term to term
 rule.
Types of sequence
There are two key types of sequences.
 
Arithmetic sequences
 
These have a term to term rule which is either an addition or a subtraction.
 
Geometric sequences
 
These have a term to term rule which is either an multiplication or a division.
Continuing sequences
For each sequence below..
a) Find the term to term rule         b) State if it is arithmetic or geometric       c) Find the next 2 terms
 
+2
 
Arithmetic
 
11, 13
 
+5
 
Arithmetic
 
25, 30
 
-1
 
Arithmetic
 
16, 15
 
x2
 
Geometric
 
80, 160
 
x10
 
Geometric
 
10000, 100000
 
+4
 
Arithmetic
 
26, 30
 
-6
 
Arithmetic
 
-9, -15
 
+0.5
 
Arithmetic
 
4, 4.5
Finding specific terms
What is the 7
th
 term in this sequence?
2, 5, 8, 11…
 
+3
 
+3
 
+3
 
+3
 
14
 
+3
 
17
 
+3
 
20
 
What is the 70
th
 term in this sequence?
 
We need a better method than just moving along the sequence!
Fortunately all sequences also have a 
position to term 
rule
 
2, 5, 8, 11, 14, 17, 20
 
3n – 1
 
The position we are looking for is ‘n’
 
So the 7
th
 term is…
 
3 x 7 - 1
 
= 20
 
And the 70
th
 term is…
 
3 x 70 - 1
 
= 209
 
We often call this the 
n
th
 term rule
.
Finding specific terms
 
5 x 
1
 + 1 =
 
6
 
5 x 
2
 + 1 =
 
11
 
5 x 
3
 + 1 =
 
16
 
5 x 
10
 + 1 =
 
51
 
5 x 
50
 + 1 =
 
251
 
30 – 2 x 
1
 =
 
28
 
30 – 2 x 
2
 =
 
26
 
30 – 2 x 
3 
=
 
24
 
30 – 2 x 
10
 =
 
10
 
30 – 2 x 
50
 =
 
-70
 
6 x 
1
 – 3 =
 
3
 
6 x 
2
 – 3 =
 
9
 
6 x 
3
 – 3 =
 
15
 
6 x 
10
 – 3 =
 
57
 
6 x 
50
 – 3 =
 
297
 
 
1
 + 8 =
 
9
 
2
 + 8 =
 
10
 
3
 + 8 =
 
11
 
10
 + 8 =
 
18
 
50
 + 8 =
 
58
Finding specific terms
Q1) What is the 4
th
 term in the sequence 3n + 6?
Q2) What is the 8
th
 term in the sequence 2n – 3?
Q3) What is the 15
th
 term in the sequence 50 – 2n?
Q4) What is the 37
th
 term in the sequence 4n – 1?
Q5) What is the 102
nd
 term in the sequence 10n – 15?
Q6) Find the first 4 terms in each of the sequences below.
a) 4n – 3
 
b) 7n + 1
 
c) n + 11
 
d) 2n – 7
 
e) 30 – 4n
 
3 x 
4
 + 6 =
 
18
 
2 x 
8
 – 3 =
 
15
 
50 – 2 x 
15
 =
 
20
 
4 x 
37
 – 1 =
 
147
 
10 x 
102
 – 15 =
 
1005
 
1, 5, 9, 13
 
8, 15, 22, 29
 
12, 13, 14, 15
 
-5, -3, -1, 1
 
26, 22, 18, 14
Finding specific terms
Checking if a number is in a sequence
Is 101 in the sequence 2n+1?
 
Position (n)
x2
+1
 
Value of the term (the sequence)
 
101
 
-1
 
÷2
 
50
 
Yes, it is the 50
th
 term in this sequence
Checking if a number is in a sequence
Is 30 in the sequence 2n+1?
 
Position (n)
x2
+1
 
Value of the term (the sequence)
 
30
 
-1
 
÷2
 
14.5
 
No, it is between the 14
th
 and 15
th
 terms
 
Is 84 in the sequence 3n + 2?
 
27.333             ÷3                  -2              84
 
No, it is not in the sequence
 
Is 84 in the sequence 5n – 1?
 
17             ÷5                  +1              84
 
Yes, it is in the sequence
 
Is 57 in the sequence 4n + 5?
 
13             ÷4                  -5              57
 
Yes, it is in the sequence
 
Is 107 in the sequence 8n – 3?
 
13.75             ÷8                  +3              107
 
No, it is not in the sequence
Checking if a number is in a sequence
1) Is 73 in the sequence 5n – 2?
2) Is 27 in the sequence 4n + 1?
3) Is 71 in the sequence 3n + 8 ?
4) Is 98 in the sequence 7n - 5?
5) What is the first number in the sequence 3n + 2 to be over 250?
6) What is the first number in the sequence 8n – 5 to be over 100?
 
+2
 
÷5
 
= 15
 
Yes, it is the 15
th
 term in the sequence
 
-1
 
÷4
 
= 6.5
 
No, it is between the 6
th
 and 7
th
 terms
 
-8
 
÷3
 
= 21
 
Yes, it is the 21
st
 term in the sequence
 
+5
 
÷7
 
= 14.714…
 
No, it is between the 14
th
 and 15
th
 terms
 
(250 – 2) ÷ 3 =
 
82.66…
 
The 83
rd
 term will be the first one that is over 250.
 
3 x 83 + 2
 
= 251
 
(100 + 5) ÷ 8 =
 
13.125
 
The 14
th
 term will be the first one that is over 100.
 
8 x 14 – 5
 
= 107
 
Year 8
 
Lockdown lessons
Challenge 1
 
x
 
-
 
2
 
6
 
5
 
4
 
x5
 
-4
 
26
 
x
 
-
 
2
 
6
 
4
 
2
 
x4
 
-2
 
22
 
a = 4
 
, b = 2
Challenge 2
 
We need to work out what position in the sequence we would find ‘0’
 
150 – 4n = 0
 
150 = 4n
 
0 is between term 37 and term 38.
 
n = 37.5
 
Term 38 will therefore be the first negative term
 
150 – 4 x 
38
 =
 
-2
Challenge 3
 
1
st
 term (n=1):
 
(1 + 1)(1 + 3)
 
= 2 x 4
 
= 8
 
2
nd
 term (n=2):
 
(2 + 1)(2 + 3)
 
= 3 x 5
 
= 15
 
3
rd
 term (n=3):
 
(3 + 1)(3 + 3)
 
= 4 x 6
 
= 24
 
= 8, 15, 24
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Delve into the world of mathematical sequences with this comprehensive guide. Explore what sequences are, different types like arithmetic and geometric, rules for continuing sequences, methods to find specific terms, and understanding position-to-term rules to solve for nth terms efficiently.


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  1. Year 8 Lockdown lessons

  2. Week 6: Lesson 1 Introduction to sequences

  3. What is a sequence? A mathematical sequence is a series of numbers connected by a rule. This rule defines their pattern 3, 8, 13, 18, 23 +5 every time 2, 4, 8, 16, 32 x2 every time -4 every time 11, 7, 3, -1, -5 Each part of a sequence is called a term How we move from one term to the next is called the term to term rule.

  4. Types of sequence There are two key types of sequences. Arithmetic sequences These have a term to term rule which is either an addition or a subtraction. Geometric sequences These have a term to term rule which is either an multiplication or a division.

  5. Continuing sequences For each sequence below.. a) Find the term to term rule b) State if it is arithmetic or geometric c) Find the next 2 terms x2 -1 +5 +2 Geometric 80, 160 Arithmetic 16, 15 Arithmetic 25, 30 Arithmetic 11, 13 +4 -6 +0.5 x10 Arithmetic 26, 30 Arithmetic -9, -15 Arithmetic 4, 4.5 Geometric 10000, 100000

  6. Finding specific terms What is the 7th term in this sequence? 2, 5, 8, 11 +3 +3 +3 +314 +317 +320 What is the 70th term in this sequence? We need a better method than just moving along the sequence!

  7. Finding specific terms Fortunately all sequences also have a position to term rule 3n 1 2, 5, 8, 11, 14, 17, 20 The position we are looking for is n So the 7thterm is = 20 3 x 7 - 1 And the 70thterm is = 209 3 x 70 - 1 We often call this the nth term rule.

  8. Finding specific terms 1st term 2nd term 3rd term 10th term 50th term Rule 5 x 50 + 1 = 251 5 x 10 + 1 = 5 x 2 + 1 = 5 x 3 + 1 = 5 x 1 + 1 = 5n + 1 51 11 16 6 30 2 x 50 = -70 30 2 x 10 = 10 30 2 x 2 = 30 2 x 3 = 30 2 x 1 = 30 2n 26 24 28 6 x 50 3 = 297 6 x 10 3 = 6 x 2 3 = 6 x 3 3 = 6 x 1 3 = 6n 3 57 9 15 3 50 + 8 = 10 + 8 = 2 + 8 = 3 + 8 = 1 + 8 = n + 8 58 18 10 11 9

  9. Finding specific terms 3 x 4 + 6 = 18 Q1) What is the 4th term in the sequence 3n + 6? 2 x 8 3 = 15 Q2) What is the 8th term in the sequence 2n 3? 50 2 x 15 = 20 Q3) What is the 15th term in the sequence 50 2n? 4 x 37 1 = 147 Q4) What is the 37th term in the sequence 4n 1? 10 x 102 15 = 1005 Q5) What is the 102nd term in the sequence 10n 15? Q6) Find the first 4 terms in each of the sequences below. e) 30 4n c) n + 11 d) 2n 7 a) 4n 3 b) 7n + 1 26, 22, 18, 14 1, 5, 9, 13 8, 15, 22, 29 12, 13, 14, 15 -5, -3, -1, 1

  10. Checking if a number is in a sequence Is 101 in the sequence 2n+1? Yes, it is the 50th term in this sequence Position (n) Value of the term (the sequence) x2 +1 50 101 2 -1

  11. Checking if a number is in a sequence Is 30 in the sequence 2n+1? No, it is between the 14th and 15th terms Position (n) Value of the term (the sequence) x2 +1 14.5 30 2 -1

  12. Is 57 in the sequence 4n + 5? Is 84 in the sequence 3n + 2? 13 4 -5 57 27.333 3 -2 84 Yes, it is in the sequence No, it is not in the sequence Is 107 in the sequence 8n 3? Is 84 in the sequence 5n 1? 13.75 8 +3 107 17 5 +1 84 No, it is not in the sequence Yes, it is in the sequence

  13. Checking if a number is in a sequence 1) Is 73 in the sequence 5n 2? +2 5 Yes, it is the 15th term in the sequence = 15 2) Is 27 in the sequence 4n + 1? -1 4 No, it is between the 6th and 7th terms = 6.5 3) Is 71 in the sequence 3n + 8 ? -8 3 Yes, it is the 21st term in the sequence = 21 4) Is 98 in the sequence 7n - 5? +5 7 No, it is between the 14th and 15th terms = 14.714 5) What is the first number in the sequence 3n + 2 to be over 250? (250 2) 3 = 82.66 The 83rd term will be the first one that is over 250. 3 x 83 + 2 = 251 6) What is the first number in the sequence 8n 5 to be over 100? The 14th term will be the first one that is over 100. (100 + 5) 8 = 13.125 8 x 14 5 = 107

  14. Year 8 Lockdown lessons

  15. Challenge 1 2 x 5 - 4 6 x5 -4 26 4 22 2 x - 6 -2 2 x4 a = 4, b = 2

  16. Challenge 2 We need to work out what position in the sequence we would find 0 150 4n = 0 0 is between term 37 and term 38. 150 = 4n n = 37.5 Term 38 will therefore be the first negative term 150 4 x 38 = -2

  17. Challenge 3 = 8 = 2 x 4 1st term (n=1): (1 + 1)(1 + 3) = 15 = 3 x 5 2nd term (n=2): (2 + 1)(2 + 3) = 24 = 4 x 6 3rd term (n=3): (3 + 1)(3 + 3) = 8, 15, 24

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