SEQUENCES AND SERIES
A sequence is an ordered list of numbers.
Note, in this context, ordered does not mean that the numbers in the list are increasing or
decreasing. Instead it means that there is a first member in the sequence, a second, a third
and so on.
The following are examples of sequences:
1, 2, 3, 4, 5,
1, 0,1, 0,1, 0,1,
1, −2, 3, −4, 5, −6,
1 1 1 1
1, , , , ,
2 3 4 5
3,1, 4,1, 5, 9, 2, 6,
A sequence can be graphed. The sequence defined by tn = n is graphed below:
The sequence defined by tn = (−1)n+1 is graphed below:
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Summer School – Unit 1 Specialist Mathematics – Book 1
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A sequence can be defined in various ways. Usually we will define sequences by:
1.
Forumula or function ( tn = f (n) )
2.
Recurrence relation, where each term is defined in terms of the previous ones (for
example, the Fibonacci sequence is often presented in this form).
In the following exercises we will see examples of both.
QUESTION 9
Write down the first four terms of the sequences defined by:
(a)
tn = 2n +1
(b)
tn = (n −1)2 + 3
(c)
tn =
1
2n +1
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(d)
tn+1 = 2tn +1, t1 = 3
(e)
tn+2 = tn+1 + tn , t1 = t2 = 1
(f)
tn+1 = ntn , t1 = −1
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ARITHMETIC SEQUENCES
An arithmetic sequence is a sequence t1, t2 , t3, which can be written in recursive form as
tn+1 = tn + d , t1 = a .
This gives us another way to write an arithmetic sequence:
a, a + d, a + 2d, a + 3d,
and so tn = a + (n −1)d .
The value of d is the common difference.
EXAMPLE 1
Determine the rule for the arithmetic sequence 3, 5, 7, 9,
Solution
The first term is 3 and the common difference is 2. The rule for the sequence is
tn = 3+ 2(n −1)
EXAMPLE 2
What is 15th term of the arithmetic sequence which has a first term of 5 and a common
difference of 3?
Solution
We have that:
t15 = 5 + (15 −1) × 3
= 5 +14 × 3
= 47
EXAMPLE 3
Determine the 20th term of the arithmetic sequence which has t2 = 10 and t10 = 66 .
Solution
We can construct simultaneous equations:
a + d = 10
a + 9d = 66
Solving these equations gives d = 7 and a = 3 . Therefore:
t20 = 3+19 × 7
= 136
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QUESTION 10
Determine the 25th term of the arithmetic sequence whose third term is 10 and whose tenth
term is 59.
Solution
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ARITHMETIC SERIES
A series is the sum of the terms of a sequence.
For the arithmetic sequence tn = a + (n −1)d , n = 1, 2, we will write the series which is the
sum of the first n terms as:
Sn = a + a + d + a + 2d ++ a + (n −1)d
It is a fairly simple exercise to show that:
Sn =
n
( 2a + (n −1)d )
2
QUESTION 11
Determine the sum of the first 20 terms of the arithmetic sequence which begins −5, −1,.
Solution
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QUESTION 12
Determine the number of terms n required to produce a sum of 60 in an arithmetic sequence
whose first term is 2 and has a common difference of 5.
Solution
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GEOMETRIC SEQUENCES
A sequence is called a geometric sequence if it is of the form:
a, ar, ar 2 , ar 3,
The first term is a and the value of r is called the common ratio. The n th term is tn = ar n−1 .
EXAMPLE 1
The sequence defined by tn = 2 n−1 , n =1, 2,3, is a geometric sequence.
The first few terms are 1, 2, 4,8,16, 32, 64,
EXAMPLE 2
1
The sequence whose n term is   , n =1, 2,3, is a geometric sequence.
2
n−1
th
1 1 1
2 4 8
The first terms of this sequence are 1, , , ,
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QUESTION 13
The numbers n − 2 , n and n + 3 are consecutive terms of a geometric sequence.
Find n and hence write down the next term.
Solution
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GEOMETRIC SERIES
We can determine the sum of the first n terms of a geometric sequence as follows:
Let Sn = a + ar + ar 2 +ar n−1
Then
rSn = ar + ar 2 ++ ar n−1 + ar n
= ( a + ar + ar 2 ++ ar n−1 ) + ar n − a
= Sn + ar n − a
Subtracting Sn from both sides and factorising gives
Sn ( r −1) = ar n − a
Sn =
a ( r n −1)
r −1
1− r n
=a
1− r
Note that
r ≠1. If r = 1 then we have a trivial geometric series and the sum Sn = na .
QUESTION 14
(a)
Determine the sum of the first 10 terms of the geometric sequence which begins
1
3, ,
3
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(b)
Determine the values of a and r if S5 = 217 and S7 = 889 and a, r > 0 .
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(c)
The 9th term of a geometric sequence is 32 times the size of the 4th term.
If the second term is 6, find the sum of the first 7 terms of the sequence.
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