S.5 Mathematics
Arithmetic and Geometric Sequences
S.5 Mathematics (Arithmetic and Geometric Sequences) Worksheet 1
＜Sequences＞
A group of numbers ordered (排序) in a specified way is called a sequence (數列).
e.g. 2, 4, 6, 8, ….
Each number in the sequence is called a term (項) of the sequence and the terms are usually
denoted by
T (1), T (2), T (3),  , T (n), 
where T (1) is the first terms, T ( 2) is the second term, T (3) is the third term, and so on,
of the sequence. T (n ) is the nth term, called the general term (通項).
Example 1
Write down the first four terms of the following sequences.
(a) T n   1n
(b) T n  2n 3  n
Example 2
Given the general term of the sequence, write down the first, fifth and the eighth terms.
(a) T n  5n  3
(b) T n  
2n  1
n2
(c) T n  5   1n1
First term
Fifth term
Eighth term
___________
___________
___________
___________
___________
___________
___________
___________
___________
Example 3
Guess the general term of the following sequences.
General Term
(a) 15, 25, 35, 45, ……
(b) 1,
2,
3 , 2,
5 , ……
(c) 39, 33, 27, 21, ……
_________________
_________________
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S.5 Mathematics
Arithmetic and Geometric Sequences
S.5 Mathematics (Arithmetic and Geometric Sequences) Worksheet 2
＜Arithmetic Sequences＞
An arithmetic sequences is a sequence having the same difference (common difference)
between any two consecutive terms (連續項).
In general, the first term (首項) of an arithmetic sequence is denoted by
common difference (公差) by d.
a
and the
Example 1
Determine whether the following sequences are arithmetic sequence. Put a「√」or「」 in the
blank.
(a)
1, 4, 9, 16, …
_______
(b)
4, 8, 12, 16, …
_______
(c)
1 2
1 2
, , 1, 1 , 1 , ...
3 3
3 3
_______
(d)
1
 4,  2,  1,  , ...
2
_______
(e)
1, 2, 1, 2, 1, 2, …
_______
A.
The General Term of an Arithmetic Sequence
Example : Find the general term of an A.S. : 2, 5, 8, 11,  .
Example : Find the general term of an A.S. : 15, 11, 7, 3,  .
An A.S. with a be the first term and d be the common difference can be written as:
a, a  d , a  2d , a  3d , .
The nth term (called the general term) of the sequence, denoted by T(n), is given by:
T (n)  a  (n  1)d
……………………(1)
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S.5 Mathematics
Arithmetic and Geometric Sequences
Example 2
If the fifth term of the arithmetic sequence is 18 and the tenth term is 58,
(a) Find the first term (a), the common difference (d) and the general term T n .
(b) If the kth term is 194, find the value of k.
(c) Is 100 a term in the sequence?
Example 3
Find the general term of an A.S. : x,  2 x,  5 x, ... .
Example 4
If
84, 75, 66, … is an arithmetic sequence, find
(a) the first positive term, ( You can find the last negative term.)
(b) the smallest term larger than 100.
Example 5
How many terms are there in the arithmetic sequence 24, 33, 42, ……, 249.
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S.5 Mathematics
B.
Arithmetic and Geometric Sequences
Arithmetic Mean (等差中項)
In an arithmetic sequence, all the intermediate terms (中間項) between two specified terms
are called the arithmetic means between the two terms.
e.g. In the arithmetic sequence: 2, 5, 8, 11, 14, 17, …,
8, 11 and 14 are the 3 arithmetic means between 5 and 17.
In particular, if a, b, c are in arithmetic sequence, the arithmetic mean
b
between
a
and
b
c
satisfies
ac
2
……………………(2)
Example 6
(a) Insert 4 arithmetic means between 32 and 7.
(b) In the arithmetic mean of (k + 3) and (2k + 6) is (2k + 2), find the value of k.
Example 7
In the figure, the widths of the highest and lowest steps of the ladder are 35cm and 89cm
respectively. In between them, there are 8 steps and they form an arithmetic sequence. Find the
widths of all the steps of this ladder.
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S.5 Mathematics
Arithmetic and Geometric Sequences
S.5 Mathematics (Arithmetic and Geometric Sequences) Worksheet 3
＜Geometric Sequences＞
A geometric sequence is a sequence having the same ratio (common ratio) between
consecutive terms.
In general, the first term (首項) of a sequence is denoted by a
and the common ratio
(公比) by r.
Example 1
Determine whether the following sequences are geometric sequence. Put a「√」or「」 in the
blank.
(a)
1, 5, 8, 11, …
_______
(b)
54, 36, 24, 16, …
_______
(c)
1,
1 1 1 1
, , , , ...
2 4 8 16
_______
(d)
1,  1, 1,  1,１...
_______
(e)
1  2, 1  2 2, 1  3 2 , …
_______
A.
The general term of a geometric sequence
Example : Find the general term of a G.S. : 2,  4, 8,  16,  .
5 5
Example : Find the general term of a G.S. : 15, 5, , ,  .
3 9
A G.S. with a be the first term and r be the common ratio can be written as:
a, ar , ar 2 , ar 3 , .
The nth term (called the general term) of the sequence, denoted by T(n), is given by:
T (n)  ar n1
……………………(3)
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S.5 Mathematics
Arithmetic and Geometric Sequences
Example 2
Given that the 3rd term and the 6th term of a geometric sequence are
3
81
and 
respectively.
4
32
(a) Find the general term of this sequence.
(b) Find T 20 and T 2k  .
Example 3
If x + 1, 6, 9
form a geometric sequence, find the value of x.
Example 4
T1 , T2 , T3 , ... form a G.S. If T1  12 and T4 : T7  3 : 2 , find T10 .
Example 5
In a geometric sequence
8
, 4, ...... , find them which is just greater than 170.
7
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S.5 Mathematics
B.
Arithmetic and Geometric Sequences
Geometric Mean (等比中項)
In a geometric sequence, all the intermediate terms between two specified terms are called
the geometric means between the two terms.
e.g. In the geometric sequence:  3,  6,  12,  24,  48,  96,  ,
 6 ,  12 ,  24 , and  48 are the 4 geometric means between  3 and  96 .
In particular, if p, q, r are in geometric sequence, the geometric mean
between
p
and
r
satisfies
q   pr
……………………(4)
Example 6
Insert 5 geometric means between 2 and
81
.
128
Example 7
If 64, a, b, c,
1
4
form a geometric sequence, find the value of
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a + b + c.
q