Download sequence

Introducing sequences
In maths, we call a list of numbers in order a sequence.
Each number in a sequence is called a term.
4, 8, 12, 16, 20, 24, 28, 32, . . .
1st term
6th term
If terms are next to each other they are referred to as
consecutive terms.
When we write out sequences, consecutive terms are
usually separated by commas.
Infinite and finite sequences
A sequence can be infinite. That means it continues forever.
For example, the sequence of multiples of 10,
10, 20 ,30, 40, 50, 60, 70, 80, 90 . . .
is infinite. We show this by adding three dots at the end.
If a sequence has a fixed number of terms it is called a
finite sequence.
For example, the sequence of two-digit square numbers
16, 25 ,36, 49, 64, 81
is finite.
Sequences and rules
Some sequences follow a simple rule that is easy to describe.
For example, this sequence
2, 5, 8, 11, 14, 17, 20, 23, 26, 29, …
continues by adding 3 each time.
In maths we are mainly concerned with sequences of
numbers that follow a rule.
Naming sequences
Here are the names of some sequences which you may
know already:
1, 3, 5, 7, 9, . . .
Odd numbers
3, 6, 9, 12, 15, . . .
Multiples of 3
5, 10, 15, 20, 25 . . .
Multiples of 5
1, 4, 9, 16, 25, . . .
Square numbers
Ascending sequences
When each term in a sequence is bigger than the one
before the sequence is called an ascending sequence.
For example,
The terms in this ascending sequence increase in equal
steps by adding 5 each time.
2,
7,
12, 17, 22, 27, 32, 37, . . .
+5
+5
+5
+5
+5
+5
+5
The terms in this ascending sequence increase in unequal
steps by starting at 0.1 and doubling each time.
0.1, 0.2, 0.4, 0.8, 1.6, 3.2, 6.4, 12.8, . . .
×2
×2
×2
×2
×2
×2
×2
Sequences that decrease in equal steps
Can you work out the next three terms in this sequence?
22,
–6
16,
10,
–6
–2,
4,
–6
–6
–8, –14, –20, . . .
–6
–6
–6
How did you work these out?
This sequence starts with 22 and decreases by 6 each time.
Sequences that increase or decrease in equal steps
are called linear or arithmetic sequences.
Sequences that increase in increasing steps
Some sequences increase or decrease in unequal steps.
For example, look at the differences between terms in this
sequence:
5,
6,
+1
8,
+2
11,
+3
15,
+4
20,
+5
26,
+6
33, . . .
+7
This sequence starts with 5 and increases by 1, 2, 3, 4, …
The differences between the terms form a linear sequence.
Sequences that decrease in decreasing steps
Can you work out the next three terms in this sequence?
7,
6.9,
–0.1
6.7,
–0.2
6.4,
–0.3
6,
–0.4
5.5,
–0.5
4.9, 4.2, . . .
–0.6
–0.7
How did you work these out?
This sequence starts with 7 and decreases by 0.1, 0.2, 0.3,
0.4, 0.5, …
With sequences of this type it is often helpful to find a second
row of differences.
Using a second row of differences
Can you work out the next three terms in this sequence?
1,
3,
+2
8,
+5
+3
16,
+8
+3
27,
+11
+3
41,
+14
+3
58,
+17
+3
78, . . .
+20
+3
Look at the differences between terms.
A sequence is formed by the differences so we look at the
second row of differences.
This shows that the differences increase by 3 each time.
Fibonacci-type sequences
Can you work out the next three terms in this sequence?
1,
1,
2,
3,
5,
8,
13,
21,
34,
1+1
1+2
3+5
5+8
8+13
13+21
21+13
55, . . .
21+34
How did you work these out?
This sequence starts 1, 1 and each term is found by
adding together the two previous terms.
This sequence is called the Fibonacci sequence after the
Italian mathematician who first wrote about it.
Writing sequences from term-to-term-rules
A term-to-term rule gives a rule for finding each term of
a sequence from the previous term or terms.
To generate a sequence from a term-to-term rule we must
also be given the first number in the sequence.
For example,
1st term
Term-to-term rule
5
Add consecutive even numbers starting with 2.
This gives us the sequence,
5
7
+2
11
+4
17
+6
27
+10
39
+12
53 . . .
+14
Sequences from a term-to-term rule
Write the first five terms of each sequence
given the first term and the term-to-term rule.
1st term
10
100
3
Term-to-term rule
Add 3
10,
13,
16,
Subtract 5
100,
95,
90,
Double
3, 6, 12,
24,
19,
21
85,
48
80
Sequences from position-to-term rules
Sometimes sequences are arranged in a table like this:
Position
1st
2nd
3rd
4th
5th
6th
…
nth
Term
3
6
9
12
15
18
…
3n
We can say that each term can be found by multiplying
the position of the term by 3.
This is called a position-to-term rule.
For this sequence we can say that the nth term is 3n,
where n is a term’s position in the sequence.
What is the 100th term in this sequence? 3 × 100 = 300
Writing sequences from position-to-term rules
The position-to-term rule for a sequence is very useful
because it allows us to work out any term in the sequence
without having to work out any other terms.
We can use algebraic shorthand to do this.
We call the first term T(1), for Term number 1,
we call the second term T(2),
we call the third term T(3), . . .
we call the nth term T(n).
T(n) is called the the nth term or the general term.
Writing sequences from position-to-term rules
For example, suppose the nth term of a sequence is 4n + 1.
We can write this rule as:
T(n) = 4n + 1
Find the first 5 terms.
T(1) = 4 × 1 + 1 = 5
T(2) = 4 × 2 + 1 = 9
T(3) = 4 × 3 + 1 = 13
T(4) = 4 × 4 + 1 = 17
T(5) = 4 × 5 + 1 = 21
The first 5 terms in the sequence are: 5, 9, 13, 17 and 21.
Writing sequences from position-to-term rules
If the nth term of a sequence is 2n2 + 3.
We can write this rule as:
T(n) = 2n2 + 3
Find the first 4 terms.
T(1) = 2 × 12 + 3 = 5
T(2) = 2 × 22 + 3 = 11
T(3) = 2 × 32 + 3 = 21
T(4) = 2 × 42 + 3 = 35
The first 4 terms in the sequence are: 5, 11, 21, and 35.
This sequence is a quadratic sequence.