The document discusses the concept of sequences and series, highlighting their applications in various fields such as physics and computer science. It explains different types of sequences, including arithmetic, geometric, and harmonic progressions, along with their means and provides examples for clarity. Additionally, it distinguishes between sequences and series, emphasizing the significance of order in sequences and summation in series.

1. Introduction
The sequence and series have important applications in Physics, Geography, Biology, Economics,
Psychology, Engineering, Chemistry, Computer science and Finance.
Sequence
A sequence is an arrangement of any objects or a set of numbers in a particular order followed by
some rule. If a1, a2, a3, a4, ……… etc. Denote the terms of a sequence, then 1,2,3,4, …
Example:
25,21,17,13,9, ….
𝑎1 = 25
𝑎2 = 25 + (−4) = 21
𝑎3 = 21 + (−4) = 17
𝑎4 = 17 + (−4) = 13
𝑎5 = 13 + (−4) = 9
𝑎6 = 9 + (−4) = 5
𝑎7 = 5 + (−4) = 1
Hence Next two terms are: 5 & 1
Sequence in computer science (programming)
 In Programming, sequence is the basic algorithm: A set of logical steps carried out in order
that commands are executed by a computer, allows us to carry out tasks that have multiple
steps.
 Microprocessor also follow a sequence which is Fetch, Decode, and then Execute.

2. Topics: -
 Arithmetic Progression (A.P)
 Arithmetic Mean (A.M)
 Geometric Progression (G.P)
 Geometric Mean (G.M)
 Harmonic Progression (H.P)
 Harmonic Mean (H.M)
Arithmetic Progression
An Arithmetic Progression is a sequence in which every term is obtained by adding some fix
number to the preceding term. The fix number to preceding term. The fix number is the common
difference of two consecutive terms denoted by d.
Example:
Q: Find the 10th term if 𝑎3 = 8 and 𝑎7 = 16
Sol: Since 𝑎3 = 8
For 3rd term:
𝑎1 + 2𝑑 = 8 ----------(1)
Also 𝑎7 = 16
𝑎1 + 6𝑑 = 16 ----------(2)
Subtract eq (1) from (2), we get
𝑎1+6𝑑 =16
𝑎1+2𝑑=8
4𝑑=8
(Subtract)
𝑑 = 2
Put 𝑑 = 2 in eq (1)
𝑎1 + 2(2) = 8
𝑎1 + 4 = 8
𝑎1 = 8 − 4
𝑎1 = 4

3. Now
𝑎10 = 𝑎1 + 9𝑑
𝑎10 = 4 + 9(2)
𝑎10 = 4 + 18
𝑎10 = 22
Arithmetic Mean
Let a and b be two numbers, than a number A is set to be A.M between a and b if a, A, b are in
A.P
Thus A - a = b – A
2A =a +b
A = a+b/2
Example:
Q: Find A.M between
2𝑥 + 1, 4𝑥 + 3
Sol:
2𝑥 + 1, 4𝑥 + 3
Here
𝑎 = 2𝑥 + 1 , 𝑏 = 4𝑥 + 3
𝐴 =
𝑎+𝑏
2
𝐴 =
2𝑥+1+4𝑥=3
2
𝐴 =
6𝑥+4
2
𝐴 =
2(3𝑥+3)
2
𝐴 = 3𝑥 + 2

4. Geometric Progression
A sequence in which every term is obtained by multiplying or dividing a definite number with the
preceding number is known as a geometric sequence.
𝑟 =
𝑎 𝑛
𝑎 𝑛−1
( 𝑛 > 1), ∀𝑛 ∈ 𝑁, 𝑎 𝑛−1 ≠ 0
Example:
Q: Find 𝑎 𝑛 if 𝑎4 =
8
27
and 𝑎7 −
64
729
of a G.P.
Sol:
Since 𝑎4 =
8
27
𝑎1 𝑟3
=
8
27
----------(1)
Also 𝑎7 = −
64
729
𝑎1 𝑟6
= −
64
729
----------(2)
Dividing eq (1) from (2), we get
𝑎1 𝑟6
𝑎1 𝑟3 =
−
644
729
8
27
𝑟3
= −
64
729
×
27
8
𝑟3
= − (
2
3
)
3
, real root only.
𝑟 = −
2
3
Put 𝑟 = −
2
3
in eq (1)
𝑎1 [−
2
3
]
3
=
8
27
−
8
27
𝑎1 =
8
27
𝑎1 = −1
𝑎 𝑛 = 𝑎1 𝑟 𝑛−1
𝑎 𝑛 = (−1)[−
2
3
]
𝑛−1
𝑎 𝑛 = (−1) 𝑛
[
2
3
]
𝑛−1

5. Geometric Mean
Let a and b be two numbers. A number G is said to be geometric mean between two numbers a
and b if a, G, b are in G.P, so
𝐺
𝑎
=
𝑏
𝐺
𝐺2
= 𝑎𝑏
𝐺 = ±𝑎𝑏
Example:
Q: Find G.M between
2i and -4i
Sol:
Here 𝑎 = 2𝑖, 𝑏 = −4𝑖
𝐺 = ±√ 𝑎𝑏
𝐺 = ±√(2𝑖)(−4𝑖)
𝐺 = ±√−8𝑖2
𝐺 = ±2√2
Harmonic Progression
If the reciprocal of the terms of the sequence form an A.P, then sequence is called harmonic.
Example:
Q: Find the 8th term of H.P:
1
2
,
1
5
,
1
8
, …
Sol:
Since
1
2
,
1
5
,
1
8
, … are in H.P

6. Therefore 2,5,8,… are in A.P
𝑎1 = 2, 𝑑 = 2
𝑎8 = 𝑎1 + 7𝑑
𝑎8 = 2 + 7(3)
𝑎8 = 2 + 21
𝑎8 = 23
In H.P
𝑎8 =
1
23
Harmonic Mean
Let a and b be two numbers, then a number H is called Harmonic mean between two numbers a
and b if a,H,b are in H.P
Example
𝐻 =
2𝑎𝑏
𝑎 + 𝑏
16 =
2𝑎(4)
𝑎 + 4
16( 𝑎 + 4) = 8𝑎
16𝑎 + 64 = 8𝑎
16𝑎 − 8𝑎 = −64
8𝑎 = −64
𝑎 = −8

7. Uses of Sequence
 A sequence is a discrete structure used to represent an ordered list.
 A sequence is a function from a subset of the set of integers (usually either the set {0,1,2..
..} Or {1,2, 3,...}To a set S.
 We use the notation an to denote the image of the integer n. ...
 Notation to represent sequence is {an}
Advantages:
The common 2-year sequence works well for many disciplines. Topics can be introduced "just-
in-time" for many disciplines. Since all students take the same sequence, advising is relatively
easy.
Difference Between Sequences and Series
Sequences Series
Set of elements that follow a pattern Sum of elements of the sequence
Order of elements is important Order of elements is not so important
Finite sequence: 1,2,3,4,5 Finite series: 1+2+3+4+5
Infinite sequence: 1,2,3,4,…… Infinite Series: 1+2+3+4+……