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SEQUENCES
Higher Tier
SEQUENCES

A sequence is a list of numbers usually connected by a rule or relationship.

Each value in a sequence is called a term.
EXAMPLE
1, 4, 7, 10, …
is a sequence
The first term ( n = 1) is 1
The second term (n = 2) is 4
etc…
The numbers are connected by the rule
add 3 or difference is 3
IMPORTANT NUMBER SEQUENCES OR NUMBER PATTERNS

You should know the following sequences:
Even Numbers:
2, 4, 6, 8, 10, …
Odd Numbers:
1, 3, 5, 7, 9, …
Square Numbers:
1, 4, 9, 16, 25, …
Cube Numbers:
1, 8, 27, 64, 125, …
Triangular Numbers:
1, 3, 6, 10, 15, …
The Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, …
RECOGNISING RULES – THE NTH TERM

Sometimes a rule can be recognized and letters used to express that rule
Example
This is the sequence of square numbers:
The nth term of this sequence is:
1, 4, 9, 16, …
n².
What is the nth term of this sequence:
2, 5, 10, 17, …
Compare the two sequences term by term.
Notice that each term in the second list is one more than the term in the first list.
The nth term is n² + 1.
sequences
©RSH 27-Mar-10
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SEQUENCES
Higher Tier
FINDING THE NTH TERM OF A LINEAR SEQUENCE

Finding the nth term means finding the rule and using letters to express that rule.

The nth term can be used to find any term in the sequence.
EXAMPLES
1.
Find the nth term of the sequence 2, 5, 8, 11, …
Solution
The common difference is 3, so the nth term will be 3n + b
The first term is 2 so with n = 1, 3  1 + b = 2.
This gives b = 1, so the nth term = 3n  1.
The 4th term (n = 4) = 3  4  1 = 11.
Check:
2.
a.
Find the nth term of the sequence 3, 9, 15, 21, …
b.
Use this rule to find the 20th term.
c.
Which term of the sequence is 147?
Solution
a.
Common difference is 6.
nth term will be 6n + b.
With n = 1, 6  1 + b = 3,  b = 3
The nth term = 6n  3.
Check:
The 3rd term (n = 3) = 6  3  3 = 15.
b.
20th term = 6  20  3 = 120  3 = 117
c.
147 = 6n  3
6n = 150
n = 25
The 25th term is 147
sequences
©RSH 27-Mar-10
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SEQUENCES
Higher Tier
EXERCISE
1.
Find an expression for the nth term of each of the following sequences:
a. 2, 4, 6, 8, …
b. 7, 9 , 11, 13, …
c. 0, 3, 6, 9, 12, …
d. 1, 4, 9, 16, …
e. 3, 6, 11, 18, …
f.
8, -2, -12, …
g. 4, 8, 12, 16, …
2.
1, 1, 2, 3, 5, 8, …
This is an example of a Fibonacci sequence.
a. Write down the next three terms in this sequence.
b. Form a new Fibonacci sequence starting with 2, 5, …
3.
a. How many squares are there in the 4th shape of this sequence.
b. Write down an expression for the nth term of this sequence.
c. Use your expression to find the number of squares in the 20 th pattern.
4.
Write down the first four terms of the sequence given by the formula
nth term = 5n  2.
What is the 18th term of this sequence?
Which term of the sequence is 63?
sequences
©RSH 27-Mar-10
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SEQUENCES
Higher Tier
QUADRATIC SEQUENCES

Rules for some quadratic sequences are easy to spot.
1, 4, 9, 16, 25, …

nth term = n²
Others can be found by comparing with known sequences.
2, 5, 10, 17, 26, …
nth term = n² + 1
2, 8, 18, 32, 50, …
nth term = 2n²
DIFFERENCES
Consider the quadratic sequence nth term = n².
Terms
1
1st difference
4
3
2nd difference
9
5
2
16
7
2
25
9
2
Consider the quadratic sequence nth term = 2n² + 1
Terms
1st difference
3
9
6
2nd difference
19
10
4
33
14
4
51
18
4

In all quadratic sequences, the 2nd difference is a constant (the same).

This fact can help establish rules for all quadratic sequences.
General Case

nth term = an² + bn + c
Terms
1st difference
a+b+c
4a + 2b + c
3a + b
5a + b
2nd difference
2a

In each case, the 2nd difference is 2a.

This fact can be used to find a, then b then c.
sequences
9a + 3b + c
©RSH 27-Mar-10
7a + b
2a
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SEQUENCES
Higher Tier
Example
The following sequence is quadratic:
8, 22, 42, 68, …
Find the rule.
Solution
Assume that the rule is an² +bn + c
Terms
8
1st difference
2nd difference
22
14
42
20
68
26
6
6
6
Then
2a = 6 which gives a = 3
And
3a + b = 14
9 + b = 14
b = 5
And
a+b+c =8
3 + 5 + c = 8
c = 0
The rule is therefore: nth term = 3n² + 5n
Check:
n = 1, 1st term = 3(1)² + 5(1) = 8
n = 2, 2nd term = 3(2)² + 5(2) = 22
n = 3, 3rd term = 3(3)² + 5(3) = 42
Examples
Sequence
Rule
5, 14, 27, 44, 65, 90, …
2n² + 3n
5, 11, 21, 35, …
2n² + 3
8, 22, 42, 68, …
3n² + 5n
3, 9, 19, 33, 51, …
2n² + 1
1, 0, 3, 8, 15, …
n²  2n
2, 7, 22, 43, 70, …
3n²  5
4, 11, 22, 37, 56, …
2n² + n + 1
sequences
©RSH 27-Mar-10
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