ICAI Notes- Sequence and Series - Arithmetic and Geometric Progressions | Quantitative Aptitude for CA Foundation PDF Download

 Page 1


SEQUENCE AND SERIES-ARITHMETIC
AND GEOMETRIC PROGRESSIONS
6
CHAPTER
Often students will come across a sequence of numbers which are having a common difference,
i.e., difference between the two consecutive pairs are the same. Also another very common
sequence of numbers which are having common ratio, i.e., ratio of two consecutive pairs are the
same. Could you guess what these special type of sequences are termed in mathematics?
Read this chapter to understand that these two special type of sequences are called Arithmetic
Progression and Geometric Progression respectively. Further learn how to find out an element
of these special sequences and how to find sum of these sequences.
These sequences will be useful for understanding various formulae of accounting and finance.
The topics of sequence, series, A.P.  G.P. find useful applications in commercial problems among
others; viz., to find interest earned through compound interest, depreciations after certain amount
of time and total sum earned on recurring deposits, etc.
S equen c e
Arithmetic Progression
Sum of n
terms
First
Term
Common
difference
Sum of first `n’
terms of the series
Sum of
n terms
First term Common
difference
Applications
of Finance
Special Series Geometric Progression
Sum of the squares of the
First `n’ terms of the series
Sum of the cubes of the
First `n’ terms of the series
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
Page 2


SEQUENCE AND SERIES-ARITHMETIC
AND GEOMETRIC PROGRESSIONS
6
CHAPTER
Often students will come across a sequence of numbers which are having a common difference,
i.e., difference between the two consecutive pairs are the same. Also another very common
sequence of numbers which are having common ratio, i.e., ratio of two consecutive pairs are the
same. Could you guess what these special type of sequences are termed in mathematics?
Read this chapter to understand that these two special type of sequences are called Arithmetic
Progression and Geometric Progression respectively. Further learn how to find out an element
of these special sequences and how to find sum of these sequences.
These sequences will be useful for understanding various formulae of accounting and finance.
The topics of sequence, series, A.P.  G.P. find useful applications in commercial problems among
others; viz., to find interest earned through compound interest, depreciations after certain amount
of time and total sum earned on recurring deposits, etc.
S equen c e
Arithmetic Progression
Sum of n
terms
First
Term
Common
difference
Sum of first `n’
terms of the series
Sum of
n terms
First term Common
difference
Applications
of Finance
Special Series Geometric Progression
Sum of the squares of the
First `n’ terms of the series
Sum of the cubes of the
First `n’ terms of the series
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
6. 2
Let us consider the following collection of numbers-
(1) 28 , 2, 25, 27, ————————
(2) 2 , 7, 11, 19, 31, 51, —————
(3) 1, 2, 3, 4, 5, 6, ———————
(4) 20, 18, 16, 14, 12, 10, —————
In (1) the nos. are not arranged in a particular order. In (2) the nos. are in ascending order but they
do not obey any rule or law. It is, therefore, not possible to indicate the number next to 51.
In (3) we find that by adding 1 to any number, we get the next one. Here the number next to 6 is
6 + 1 = 7.
In (4) if we subtract 2 from any number we get the nos. that follows. Here the number next to 10
is 10 –2 = 8.
Under these circumstances, we say, the numbers in the collections (1) and (2) do not form sequences
whereas the numbers in the collections (3) & (4) form sequences.
Thus a sequence may be defined as follows:—
An ordered collection of numbers a
1
, a
2
, a
3
, a
4
, ................., a
n
, ................. is a sequence if according
to some definite rule or law, there is a definite value of a
n , 
called the term or element of the
sequence, corresponding to any value of the natural number n.
Clearly, a
1 
is the 1st term of the sequence , a
2 
is the 2nd term, ................., a
n
 is the nth term.
In the nth term a
n 
, by putting n = 1, 2 ,3 ,......... successively , we get a
1
, a
2
 , a
3
 , a
4
, .........
Thus it is clear that the nth term of a sequence is a function of the positive integer n. The nth term
is also called the general term of the sequence. To specify a sequence, nth term must be known,
otherwise it may lead to confusion. A sequence may be finite or infinite.
If the number of elements in a sequence is finite, the sequence is called finite sequence; while if the
number of elements is unending, the sequence is infinite.
A finite sequence a
1
, a
2
, a
3
, a
4
, ................., a
n
 is denoted by ? ?
n
i
i=1
a and  an  infinite  sequence a
1
, a
2
,
a
3
, a
4
, ................., a
n
 
,
................. is denoted by ? ?
?
n
n=1
a or simply by
{ a
n
 } where a
n 
is the nth element of the sequence.
Example :
1) The sequence { 1/n } is 1, 1/2, 1/3, 1/4……
2) The sequence { ( – 1 ) 
n
 n } is –1, 2, –3, 4, –5,…..
3) The sequence { n } is 1, 2, 3,…
4) The sequence { n / (n + 1) } is 1/2, 2/3, 3/4, 4/5 …….
5) A sequence of even positive integers is 2, 4, 6, .....................................
6) A sequence of odd positive integers is 1, 3, 5, 7, .....................................
All the above are infinite sequences.
© The Institute of Chartered Accountants of India
Page 3


SEQUENCE AND SERIES-ARITHMETIC
AND GEOMETRIC PROGRESSIONS
6
CHAPTER
Often students will come across a sequence of numbers which are having a common difference,
i.e., difference between the two consecutive pairs are the same. Also another very common
sequence of numbers which are having common ratio, i.e., ratio of two consecutive pairs are the
same. Could you guess what these special type of sequences are termed in mathematics?
Read this chapter to understand that these two special type of sequences are called Arithmetic
Progression and Geometric Progression respectively. Further learn how to find out an element
of these special sequences and how to find sum of these sequences.
These sequences will be useful for understanding various formulae of accounting and finance.
The topics of sequence, series, A.P.  G.P. find useful applications in commercial problems among
others; viz., to find interest earned through compound interest, depreciations after certain amount
of time and total sum earned on recurring deposits, etc.
S equen c e
Arithmetic Progression
Sum of n
terms
First
Term
Common
difference
Sum of first `n’
terms of the series
Sum of
n terms
First term Common
difference
Applications
of Finance
Special Series Geometric Progression
Sum of the squares of the
First `n’ terms of the series
Sum of the cubes of the
First `n’ terms of the series
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
6. 2
Let us consider the following collection of numbers-
(1) 28 , 2, 25, 27, ————————
(2) 2 , 7, 11, 19, 31, 51, —————
(3) 1, 2, 3, 4, 5, 6, ———————
(4) 20, 18, 16, 14, 12, 10, —————
In (1) the nos. are not arranged in a particular order. In (2) the nos. are in ascending order but they
do not obey any rule or law. It is, therefore, not possible to indicate the number next to 51.
In (3) we find that by adding 1 to any number, we get the next one. Here the number next to 6 is
6 + 1 = 7.
In (4) if we subtract 2 from any number we get the nos. that follows. Here the number next to 10
is 10 –2 = 8.
Under these circumstances, we say, the numbers in the collections (1) and (2) do not form sequences
whereas the numbers in the collections (3) & (4) form sequences.
Thus a sequence may be defined as follows:—
An ordered collection of numbers a
1
, a
2
, a
3
, a
4
, ................., a
n
, ................. is a sequence if according
to some definite rule or law, there is a definite value of a
n , 
called the term or element of the
sequence, corresponding to any value of the natural number n.
Clearly, a
1 
is the 1st term of the sequence , a
2 
is the 2nd term, ................., a
n
 is the nth term.
In the nth term a
n 
, by putting n = 1, 2 ,3 ,......... successively , we get a
1
, a
2
 , a
3
 , a
4
, .........
Thus it is clear that the nth term of a sequence is a function of the positive integer n. The nth term
is also called the general term of the sequence. To specify a sequence, nth term must be known,
otherwise it may lead to confusion. A sequence may be finite or infinite.
If the number of elements in a sequence is finite, the sequence is called finite sequence; while if the
number of elements is unending, the sequence is infinite.
A finite sequence a
1
, a
2
, a
3
, a
4
, ................., a
n
 is denoted by ? ?
n
i
i=1
a and  an  infinite  sequence a
1
, a
2
,
a
3
, a
4
, ................., a
n
 
,
................. is denoted by ? ?
?
n
n=1
a or simply by
{ a
n
 } where a
n 
is the nth element of the sequence.
Example :
1) The sequence { 1/n } is 1, 1/2, 1/3, 1/4……
2) The sequence { ( – 1 ) 
n
 n } is –1, 2, –3, 4, –5,…..
3) The sequence { n } is 1, 2, 3,…
4) The sequence { n / (n + 1) } is 1/2, 2/3, 3/4, 4/5 …….
5) A sequence of even positive integers is 2, 4, 6, .....................................
6) A sequence of odd positive integers is 1, 3, 5, 7, .....................................
All the above are infinite sequences.
© The Institute of Chartered Accountants of India
6. 3 SEQUENCE AND SERIES-ARITHMETIC AND GEOMETRIC PROGRESSIONS
Example:
1) A sequence of even positive integers within 12 i.e., is 2, 4, 6, 8, 10.
2) A sequence of odd positive integers within 11 i.e., is 1, 3, 5, 7, 9.
All the above are finite sequences.
An expression of the form a
1
 + a
2
 + a
3
 + ….. + a
n
 + ............................ which is the sum of the
elements of the sequence { a } is called a 
n
series. If the series contains a finite number of elements,
it is called a finite series, otherwise called an infinite series.
If S
n
 = u
1
 + u
2
 + u
3
 + u
4
 + ……. + u
n
, then S
n
 is called the sum to n terms (or the sum of the first n
terms) of the series and the term sum is denoted by the Greek letter ?.
Thus, S
n
 = 
?
n
r=1
u
r
 or simply by ?u
n.
 ILLUSTRATIONS:
(i) 1 + 3 + 5 + 7 + ............................ is a series in which 1st term = 1, 2nd term = 3 , and so on.
(ii) 2 – 4 + 8 –16 + ..................... is also a series in which 1st term = 2, 2nd term = –4 , and so on.
A sequence a
1
, a
2
 ,a
3
, ……, a
n 
is called an Arithmetic Progression (A.P.) when a
2
 – a
1
 = a
3
 – a
2
 = …..
= a
n
 – a
n–1
. That means A. P. is a sequence in which each term is obtained by adding a constant d
to the preceding term. This constant ‘d’ is called the common difference of the A.P. If 3 numbers a,
b, c are in A.P., we say
b – a = c – b or a + c = 2b; b is called the arithmetic mean between a and c.
Example: 1) 2,5,8,11,14,17,…… is an A.P. in which d = 3 is the common diference.
2) 15,13,11,9,7,5,3,1,–1, is an A.P. in which –2 is the common difference.
Solution: In (1) 2nd term = 5 , 1st term = 2, 3rd term = 8,
so 2nd term – 1st term = 5 – 2 = 3, 3rd term – 2nd term = 8 – 5 = 3
Here the difference between a term and the preceding term is same that is always constant. This
constant is called common difference.
Now in generel an A.P. series can be written as
a, a + d, a + 2d, a + 3d, a + 4d, ……
where ‘a’ is the 1
st
 term and ‘d’ is the common difference.
Thus 1
st
 term ( t
1
 ) = a = a + ( 1 – 1 ) d
2
nd
 term ( t
2
 ) = a + d = a + ( 2 – 1 ) d
3
rd
 term (t
3
) = a + 2d = a + (3 – 1) d
© The Institute of Chartered Accountants of India
Page 4


SEQUENCE AND SERIES-ARITHMETIC
AND GEOMETRIC PROGRESSIONS
6
CHAPTER
Often students will come across a sequence of numbers which are having a common difference,
i.e., difference between the two consecutive pairs are the same. Also another very common
sequence of numbers which are having common ratio, i.e., ratio of two consecutive pairs are the
same. Could you guess what these special type of sequences are termed in mathematics?
Read this chapter to understand that these two special type of sequences are called Arithmetic
Progression and Geometric Progression respectively. Further learn how to find out an element
of these special sequences and how to find sum of these sequences.
These sequences will be useful for understanding various formulae of accounting and finance.
The topics of sequence, series, A.P.  G.P. find useful applications in commercial problems among
others; viz., to find interest earned through compound interest, depreciations after certain amount
of time and total sum earned on recurring deposits, etc.
S equen c e
Arithmetic Progression
Sum of n
terms
First
Term
Common
difference
Sum of first `n’
terms of the series
Sum of
n terms
First term Common
difference
Applications
of Finance
Special Series Geometric Progression
Sum of the squares of the
First `n’ terms of the series
Sum of the cubes of the
First `n’ terms of the series
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
6. 2
Let us consider the following collection of numbers-
(1) 28 , 2, 25, 27, ————————
(2) 2 , 7, 11, 19, 31, 51, —————
(3) 1, 2, 3, 4, 5, 6, ———————
(4) 20, 18, 16, 14, 12, 10, —————
In (1) the nos. are not arranged in a particular order. In (2) the nos. are in ascending order but they
do not obey any rule or law. It is, therefore, not possible to indicate the number next to 51.
In (3) we find that by adding 1 to any number, we get the next one. Here the number next to 6 is
6 + 1 = 7.
In (4) if we subtract 2 from any number we get the nos. that follows. Here the number next to 10
is 10 –2 = 8.
Under these circumstances, we say, the numbers in the collections (1) and (2) do not form sequences
whereas the numbers in the collections (3) & (4) form sequences.
Thus a sequence may be defined as follows:—
An ordered collection of numbers a
1
, a
2
, a
3
, a
4
, ................., a
n
, ................. is a sequence if according
to some definite rule or law, there is a definite value of a
n , 
called the term or element of the
sequence, corresponding to any value of the natural number n.
Clearly, a
1 
is the 1st term of the sequence , a
2 
is the 2nd term, ................., a
n
 is the nth term.
In the nth term a
n 
, by putting n = 1, 2 ,3 ,......... successively , we get a
1
, a
2
 , a
3
 , a
4
, .........
Thus it is clear that the nth term of a sequence is a function of the positive integer n. The nth term
is also called the general term of the sequence. To specify a sequence, nth term must be known,
otherwise it may lead to confusion. A sequence may be finite or infinite.
If the number of elements in a sequence is finite, the sequence is called finite sequence; while if the
number of elements is unending, the sequence is infinite.
A finite sequence a
1
, a
2
, a
3
, a
4
, ................., a
n
 is denoted by ? ?
n
i
i=1
a and  an  infinite  sequence a
1
, a
2
,
a
3
, a
4
, ................., a
n
 
,
................. is denoted by ? ?
?
n
n=1
a or simply by
{ a
n
 } where a
n 
is the nth element of the sequence.
Example :
1) The sequence { 1/n } is 1, 1/2, 1/3, 1/4……
2) The sequence { ( – 1 ) 
n
 n } is –1, 2, –3, 4, –5,…..
3) The sequence { n } is 1, 2, 3,…
4) The sequence { n / (n + 1) } is 1/2, 2/3, 3/4, 4/5 …….
5) A sequence of even positive integers is 2, 4, 6, .....................................
6) A sequence of odd positive integers is 1, 3, 5, 7, .....................................
All the above are infinite sequences.
© The Institute of Chartered Accountants of India
6. 3 SEQUENCE AND SERIES-ARITHMETIC AND GEOMETRIC PROGRESSIONS
Example:
1) A sequence of even positive integers within 12 i.e., is 2, 4, 6, 8, 10.
2) A sequence of odd positive integers within 11 i.e., is 1, 3, 5, 7, 9.
All the above are finite sequences.
An expression of the form a
1
 + a
2
 + a
3
 + ….. + a
n
 + ............................ which is the sum of the
elements of the sequence { a } is called a 
n
series. If the series contains a finite number of elements,
it is called a finite series, otherwise called an infinite series.
If S
n
 = u
1
 + u
2
 + u
3
 + u
4
 + ……. + u
n
, then S
n
 is called the sum to n terms (or the sum of the first n
terms) of the series and the term sum is denoted by the Greek letter ?.
Thus, S
n
 = 
?
n
r=1
u
r
 or simply by ?u
n.
 ILLUSTRATIONS:
(i) 1 + 3 + 5 + 7 + ............................ is a series in which 1st term = 1, 2nd term = 3 , and so on.
(ii) 2 – 4 + 8 –16 + ..................... is also a series in which 1st term = 2, 2nd term = –4 , and so on.
A sequence a
1
, a
2
 ,a
3
, ……, a
n 
is called an Arithmetic Progression (A.P.) when a
2
 – a
1
 = a
3
 – a
2
 = …..
= a
n
 – a
n–1
. That means A. P. is a sequence in which each term is obtained by adding a constant d
to the preceding term. This constant ‘d’ is called the common difference of the A.P. If 3 numbers a,
b, c are in A.P., we say
b – a = c – b or a + c = 2b; b is called the arithmetic mean between a and c.
Example: 1) 2,5,8,11,14,17,…… is an A.P. in which d = 3 is the common diference.
2) 15,13,11,9,7,5,3,1,–1, is an A.P. in which –2 is the common difference.
Solution: In (1) 2nd term = 5 , 1st term = 2, 3rd term = 8,
so 2nd term – 1st term = 5 – 2 = 3, 3rd term – 2nd term = 8 – 5 = 3
Here the difference between a term and the preceding term is same that is always constant. This
constant is called common difference.
Now in generel an A.P. series can be written as
a, a + d, a + 2d, a + 3d, a + 4d, ……
where ‘a’ is the 1
st
 term and ‘d’ is the common difference.
Thus 1
st
 term ( t
1
 ) = a = a + ( 1 – 1 ) d
2
nd
 term ( t
2
 ) = a + d = a + ( 2 – 1 ) d
3
rd
 term (t
3
) = a + 2d = a + (3 – 1) d
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
6 . 4
4
th
 term (t
4
) = a + 3d = a + (4 – 1) d
…………………………………………….
n
th
 term (t
n
) = a + ( n – 1) d, where n is the position number of the term.
Using this formula we can get
50
th
 term (= t
50
) = a+ (50 – 1) d = a + 49d
Example 1: Find the 7th term of the A.P. 8, 5, 2, –1, –4,…..
Solution: Here     a = 8, d = 5 – 8 = –3
Now    t
7
= 8 + (7 – 1) d
= 8 + (7 – 1) (– 3)
= 8 + 6 (– 3)
= 8 – 18
= – 10
Example 2: Which term of the AP 
3 4 5 17
, , ............is ?
7 7 7 7
Solution:  a = 
-
n
3 4 3 1 17
, d = = , t =
7 7 7 7 7
We may write
( n
17 3 1
= - 1)
7 7 7
? ?
or, 17 = 3 + ( n – 1)
or, n = 17 – 2 = 15
Hence, 15
th
 term of the A.P. is 
.
17
7
Example 3: If 5
th
 and 12
th
 terms of an A.P. are 14 and 35 respectively, find the A.P.
Solution:  Let a be the first term & d be the common difference of A.P.
t
5 
= a + 4d = 14
t
12
 = a + 11d = 35
On solving the above two equations,
7d = 21   = i.e., d = 3
and a = 14 – (4 × 3) = 14 – 12 = 2
© The Institute of Chartered Accountants of India
Page 5


SEQUENCE AND SERIES-ARITHMETIC
AND GEOMETRIC PROGRESSIONS
6
CHAPTER
Often students will come across a sequence of numbers which are having a common difference,
i.e., difference between the two consecutive pairs are the same. Also another very common
sequence of numbers which are having common ratio, i.e., ratio of two consecutive pairs are the
same. Could you guess what these special type of sequences are termed in mathematics?
Read this chapter to understand that these two special type of sequences are called Arithmetic
Progression and Geometric Progression respectively. Further learn how to find out an element
of these special sequences and how to find sum of these sequences.
These sequences will be useful for understanding various formulae of accounting and finance.
The topics of sequence, series, A.P.  G.P. find useful applications in commercial problems among
others; viz., to find interest earned through compound interest, depreciations after certain amount
of time and total sum earned on recurring deposits, etc.
S equen c e
Arithmetic Progression
Sum of n
terms
First
Term
Common
difference
Sum of first `n’
terms of the series
Sum of
n terms
First term Common
difference
Applications
of Finance
Special Series Geometric Progression
Sum of the squares of the
First `n’ terms of the series
Sum of the cubes of the
First `n’ terms of the series
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
6. 2
Let us consider the following collection of numbers-
(1) 28 , 2, 25, 27, ————————
(2) 2 , 7, 11, 19, 31, 51, —————
(3) 1, 2, 3, 4, 5, 6, ———————
(4) 20, 18, 16, 14, 12, 10, —————
In (1) the nos. are not arranged in a particular order. In (2) the nos. are in ascending order but they
do not obey any rule or law. It is, therefore, not possible to indicate the number next to 51.
In (3) we find that by adding 1 to any number, we get the next one. Here the number next to 6 is
6 + 1 = 7.
In (4) if we subtract 2 from any number we get the nos. that follows. Here the number next to 10
is 10 –2 = 8.
Under these circumstances, we say, the numbers in the collections (1) and (2) do not form sequences
whereas the numbers in the collections (3) & (4) form sequences.
Thus a sequence may be defined as follows:—
An ordered collection of numbers a
1
, a
2
, a
3
, a
4
, ................., a
n
, ................. is a sequence if according
to some definite rule or law, there is a definite value of a
n , 
called the term or element of the
sequence, corresponding to any value of the natural number n.
Clearly, a
1 
is the 1st term of the sequence , a
2 
is the 2nd term, ................., a
n
 is the nth term.
In the nth term a
n 
, by putting n = 1, 2 ,3 ,......... successively , we get a
1
, a
2
 , a
3
 , a
4
, .........
Thus it is clear that the nth term of a sequence is a function of the positive integer n. The nth term
is also called the general term of the sequence. To specify a sequence, nth term must be known,
otherwise it may lead to confusion. A sequence may be finite or infinite.
If the number of elements in a sequence is finite, the sequence is called finite sequence; while if the
number of elements is unending, the sequence is infinite.
A finite sequence a
1
, a
2
, a
3
, a
4
, ................., a
n
 is denoted by ? ?
n
i
i=1
a and  an  infinite  sequence a
1
, a
2
,
a
3
, a
4
, ................., a
n
 
,
................. is denoted by ? ?
?
n
n=1
a or simply by
{ a
n
 } where a
n 
is the nth element of the sequence.
Example :
1) The sequence { 1/n } is 1, 1/2, 1/3, 1/4……
2) The sequence { ( – 1 ) 
n
 n } is –1, 2, –3, 4, –5,…..
3) The sequence { n } is 1, 2, 3,…
4) The sequence { n / (n + 1) } is 1/2, 2/3, 3/4, 4/5 …….
5) A sequence of even positive integers is 2, 4, 6, .....................................
6) A sequence of odd positive integers is 1, 3, 5, 7, .....................................
All the above are infinite sequences.
© The Institute of Chartered Accountants of India
6. 3 SEQUENCE AND SERIES-ARITHMETIC AND GEOMETRIC PROGRESSIONS
Example:
1) A sequence of even positive integers within 12 i.e., is 2, 4, 6, 8, 10.
2) A sequence of odd positive integers within 11 i.e., is 1, 3, 5, 7, 9.
All the above are finite sequences.
An expression of the form a
1
 + a
2
 + a
3
 + ….. + a
n
 + ............................ which is the sum of the
elements of the sequence { a } is called a 
n
series. If the series contains a finite number of elements,
it is called a finite series, otherwise called an infinite series.
If S
n
 = u
1
 + u
2
 + u
3
 + u
4
 + ……. + u
n
, then S
n
 is called the sum to n terms (or the sum of the first n
terms) of the series and the term sum is denoted by the Greek letter ?.
Thus, S
n
 = 
?
n
r=1
u
r
 or simply by ?u
n.
 ILLUSTRATIONS:
(i) 1 + 3 + 5 + 7 + ............................ is a series in which 1st term = 1, 2nd term = 3 , and so on.
(ii) 2 – 4 + 8 –16 + ..................... is also a series in which 1st term = 2, 2nd term = –4 , and so on.
A sequence a
1
, a
2
 ,a
3
, ……, a
n 
is called an Arithmetic Progression (A.P.) when a
2
 – a
1
 = a
3
 – a
2
 = …..
= a
n
 – a
n–1
. That means A. P. is a sequence in which each term is obtained by adding a constant d
to the preceding term. This constant ‘d’ is called the common difference of the A.P. If 3 numbers a,
b, c are in A.P., we say
b – a = c – b or a + c = 2b; b is called the arithmetic mean between a and c.
Example: 1) 2,5,8,11,14,17,…… is an A.P. in which d = 3 is the common diference.
2) 15,13,11,9,7,5,3,1,–1, is an A.P. in which –2 is the common difference.
Solution: In (1) 2nd term = 5 , 1st term = 2, 3rd term = 8,
so 2nd term – 1st term = 5 – 2 = 3, 3rd term – 2nd term = 8 – 5 = 3
Here the difference between a term and the preceding term is same that is always constant. This
constant is called common difference.
Now in generel an A.P. series can be written as
a, a + d, a + 2d, a + 3d, a + 4d, ……
where ‘a’ is the 1
st
 term and ‘d’ is the common difference.
Thus 1
st
 term ( t
1
 ) = a = a + ( 1 – 1 ) d
2
nd
 term ( t
2
 ) = a + d = a + ( 2 – 1 ) d
3
rd
 term (t
3
) = a + 2d = a + (3 – 1) d
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
6 . 4
4
th
 term (t
4
) = a + 3d = a + (4 – 1) d
…………………………………………….
n
th
 term (t
n
) = a + ( n – 1) d, where n is the position number of the term.
Using this formula we can get
50
th
 term (= t
50
) = a+ (50 – 1) d = a + 49d
Example 1: Find the 7th term of the A.P. 8, 5, 2, –1, –4,…..
Solution: Here     a = 8, d = 5 – 8 = –3
Now    t
7
= 8 + (7 – 1) d
= 8 + (7 – 1) (– 3)
= 8 + 6 (– 3)
= 8 – 18
= – 10
Example 2: Which term of the AP 
3 4 5 17
, , ............is ?
7 7 7 7
Solution:  a = 
-
n
3 4 3 1 17
, d = = , t =
7 7 7 7 7
We may write
( n
17 3 1
= - 1)
7 7 7
? ?
or, 17 = 3 + ( n – 1)
or, n = 17 – 2 = 15
Hence, 15
th
 term of the A.P. is 
.
17
7
Example 3: If 5
th
 and 12
th
 terms of an A.P. are 14 and 35 respectively, find the A.P.
Solution:  Let a be the first term & d be the common difference of A.P.
t
5 
= a + 4d = 14
t
12
 = a + 11d = 35
On solving the above two equations,
7d = 21   = i.e., d = 3
and a = 14 – (4 × 3) = 14 – 12 = 2
© The Institute of Chartered Accountants of India
6 . 5 SEQUENCE AND SERIES-ARITHMETIC AND GEOMETRIC PROGRESSIONS
Hence, the required A.P. is 2, 5, 8, 11, 14,……………
Example 4: Divide 69 into three parts which are in A.P. and are such that the product of the first
two parts is 483.
Solution: Given that the three parts are in A.P., let the three parts which are in A.P. be a – d, a,
a + d.........
Thus a – d + a + a + d = 69
or 3a = 69
or a = 23
So the three parts are 23 – d, 23, 23 + d
Since the product of first two parts is 483, therefore, we have
23 ( 23 – d ) = 483
or 23 – d = 483 / 23 = 21
or d = 23 – 21 = 2
Hence, the three parts which are in A.P. are
23 – 2 = 21, 23, 23 + 2 = 25
Hence the three parts are 21, 23, 25.
Example 5: Find the arithmetic mean between 4 and 10.
Solution: We know that the A.M. of a & b is = ( a + b ) /2
Hence, The A. M between 4 & 10 = ( 4 + 10 ) /2 = 7
Example 6: Insert 4 arithmetic means between 4 and 324.
4, –, –, –, –, 324
Solution: Here a= 4, d = ? n = 2 + 4 = 6, t
n
 = 324
Now t
n
 = a + ( n – 1 ) d
or 324= 4 + ( 6 – 1 ) d
or 320= 5d i.e., = i.e., d = 320 / 5 = 64
So the 1
st
 AM = 4 + 64 = 68
2
nd
 AM = 68 + 64 = 132
3
rd
 AM = 132 + 64 = 196
4
th
 AM = 196 + 64 = 260
Sum of the first
 
 n terms
Let S be the Sum, a be the 1
st
 term and ? the last term of an A.P. If the number of term is n, then t
n
= ?. Let d be the common difference of the A.P.
Now S = a + ( a + d ) + ( a + 2d ) + .. + ( ? – 2d ) + ( ? – d ) + ?
Again S = ? + ( ? – d ) + ( ? – 2d ) + …. + ( a + 2d ) + ( a + d ) + a
© The Institute of Chartered Accountants of India