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Sequences
Objectives:
Grade D:
Write the terms of a sequence given the nth term
Grade C:
Find the nth term of a sequence or a series of diagrams
Sequences
Definitions:
Sequence
A list of numbers or diagrams that are connected in some way.
Term
A term is a number and / or variable(s) connected with
x and / or ÷ separated from anther term by an ‘+’ or ‘-’ operation.
nth term
The ‘general’ term used to describe a sequence, e.g. 3n + 1
If you are given the nth term you can find the terms of a
sequence.
Coefficient
The number preceding a letter – the number that is used to
multiply a letter.
e.g.
2n
the coefficient is 2
Sequences
The term to term rule is the rule that connects numbers in a sequence:
e.g. Find the term to term rule for this sequence:
5, 7, 9, 11, ...
The dots show that the sequence continues
Each term (consecutive number in the sequence) is 2 more than the
one before it, so the rule is:
+2
Sequences
Now try these:
Find the term to term rule for these sequences:
1)
3, 7, 11, 15, ...
+4
2)
0, 5, 10, 15, ...
+5
3)
16, 13, 10, 7, ...
-3
4)
3, 1, -1, - 3, ...
-2
5)
1, 2, 4, 8, 16,
x2
6)
2, 3.5, 5, 6.5, ...
+1.5
7)
0.01, 0.1, 1, 10, ...
x 10
Sequences
To write the terms of a sequence given the nth term
Given the expression:
2n + 3
write the first 5 terms
In this expression the letter n represents the term number and thus
if we substitute the term number for the letter n we will find value
that particular term.
The first 5 terms of the sequence will be using values for n of
1, 2, 3, 4 and 5
term 1
term 2
term 3
term 4
term 5
2x1+3
2x2+3
5
7
2x3+3 2x4+3
9
11
2x5+3
13
Sequences
Now try these:
Write the first 5 terms of these sequences:
1)
n+2
3, 4, 5, 6, 7
2)
2n + 5
7, 9, 11, 13, 15
3)
3n - 2
1, 4, 7, 10, 13
4)
5n + 3
8, 13, 18, 23, 28
5)
-4n
6, 2, - 2, - 6, - 10
+ 10
6)
n2 + 2
3, 6, 11, 18, 27
7)
3n2
3, 12, 27, 48, 75
Sequences
Look at the nth term
n +2 and the sequence it generates:
3, 4, 5, 6, 7
Find the term to term rule:
+1
Look at the nth term
2n +5 and the sequence it generates:
7, 9, 11, 13, 15
Find the term to term rule:
+2
You will notice that the coefficient in each expression is the
term to term rule:
n +2 rule +1
2n +5 rule +2
The coefficient of n is 1 because if we multiply n by 1 it is still n
Sequences
To find the nth term of a sequence or a series of diagrams
this shows us that the coefficient for a sequence is the term to
term rule, so we always find this first.
Example:
Find the nth term for the following sequence: 3, 7, 11, 15
The term to term rule is: +4
Therefore the coefficient of n is 4 so we write
4n
However, if we find the first 4 terms of this we get:
4, 8, 12, 16
This doesn’t give the correct sequence, but you will notice that
If you subtract 1 from each of these terms you get the correct sequence
The full nth term is therefore:
4n - 1
Sequences
To summarise finding the nth term:
• Find the term to term rule
• Find how much you need to add or subtract
to get the correct sequence
• Check your nth term works for the 2nd and 3rd terms
Try this:
Find the nth term for the following sequence:
+3
Write:
5, 8, 11, 14
3n
use n = 1 for the 1st term
the 1st term would be 3 x 1 = 3 We need to + 2 to make the correct
first term of 5
The nth term becomes 3n + 2
Check this in terms 2 and 3
3x2+2=8
3 x 3 + 2 = 11
Sequences
Now try these:
Find the nth term for these sequences:
1)
4, 6, 8, 10, ...
2n + 2
2)
1, 6, 11, 16, ...
5n - 4
3)
3, 10, 17, 24, ...
7n - 4
4)
10, 19, 28, 37, ...
9n + 1
5)
13, 10, 7, 4, ...
-3n + 16
6)
20, 14, 8, 2, ...
-6n + 26
Sequences
Find the nth term of a series of diagrams
Here is a series of diagrams
5
13
9
Write the number of matches in each pattern
Now we have a sequence of numbers
from which we can find the nth term
The term to term rule is
+4
The nth term is therefore
4n + 1
17
Sequences
Now try these:
Sequences
Now try these:
4
7
10
13
3n + 1
6
10
14
18
4n + 2
16
28
40
12n + 4