Sequences and Series
Indrottama
September 2023
1 Arithmetic Progression
1.1 Introduction
An arithmetic progression (AP) is a sequence of numbers in which the difference
between any two consecutive terms is constant. This constant difference is
called the common difference, denoted by d. The general form of an arithmetic
progression is:
a, a + d, a + 2d, a + 3d, . . .
Where:
a : The first term of the AP
d : The common difference
In this section, we will explore the formulas and properties of arithmetic
progressions.
1.2 Formulas
1.2.1 Formula for the nth Term
The formula to find the nth term an of an arithmetic progression is given by:
an = a + (n − 1)d
1.2.2 Formula for the Sum of n Terms
The sum of the first n terms of an arithmetic progression, denoted by Sn , can
be calculated using the formula:
n
Sn = [2a + (n − 1)d]
2
1
1.2.3 Number of Terms in an Arithmetic Progression
To find the number of terms n in an arithmetic progression when you know
the first term a, the last term l, and the common difference d, you can use the
formula:
l−a
n= +1
d
1.3 Conclusion
Arithmetic progressions are a fundamental concept in mathematics. They con-
sist of a sequence of numbers with a constant difference between consecutive
terms. The key formulas and properties discussed in this section can be used to
calculate specific terms or the sum of terms in an arithmetic progression, making
them valuable tools for various mathematical and practical applications.
2 Harmonic Progression
2.1 Introduction
A harmonic progression (HP) is a sequence of numbers in which the reciprocals
of the terms form an arithmetic progression. In other words, in a harmonic
progression, the reciprocals of the terms have a constant difference. The general
form of a harmonic progression is:
1 1 1 1
, , , ,...
a a + d a + 2d a + 3d
Where:
a : The first term of the HP
d : The common difference of reciprocals
In this section, we will explore the formulas and properties of harmonic
progressions.
2.2 Formulas
2.2.1 Formula for the nth Term
1
The formula to find the nth term of a harmonic progression is given by:
hn
1 1 1
= + (n − 1)
hn a d
This can be simplified as:
1
hn = 1 1
a + (n − 1) d
2
2.2.2 Formula for the Sum of n Terms
The sum of the first n terms of a harmonic progression, denoted by Hn , can be
calculated using the formula:
n (n − 1)n
Hn = +
a 2d
2.2.3 Number of Terms in a Harmonic Progression
To find the number of terms n in a harmonic progression when you know the
first term a1 , the last term 1l , and the common difference of reciprocals d1 , you
can use the formula:
1 1
l − a
n= 1 +1
d
2.3 Conclusion
Harmonic progressions are a unique type of sequence where the reciprocals of
terms form an arithmetic progression. The formulas and properties discussed
in this section can be used to calculate specific terms or the sum of terms in
a harmonic progression, providing valuable tools for various mathematical and
practical applications.
3 Geometric Series
A geometric series is a special type of series in the field of number sequences. It
is characterized by a sequence of numbers where each term is obtained by multi-
plying the previous term by a fixed number called the common ratio. Geometric
series are also known as geometric progressions (abbreviated as G.P.).
The general form of a geometric series is as follows:
a, ar, ar2 , ar3 , . . . , arn−1 , . . .
In this sequence:
a : The first term of the geometric series
r : The common ratio
n : The number of terms
Note: The common ratio, r, in a geometric series is a positive or negative
integer but cannot be equal to 1.
3
3.1 Geometric Series Formula
The formulas for geometric series encompass the calculation of the sum of a finite
geometric series, the sum of an infinite geometric series, and the determination
of the nth term of a geometric series.
3.1.1 Recursive Formula of Geometric Series
Given a geometric series with the common ratio r, the recursive formula for the
series is as follows:
an = ran−1 , for n ≥ 2
Here, an represents the nth term, and an−1 represents the (n-1)th term in the
series.
3.1.2 Geometric Series Formula for the nth Term
The formula to find the nth (or general) term of a geometric series, given the
first term a1 and the common ratio r, is as follows:
an = a1 · rn−1
3.1.3 Sum of Finite Geometric Series Formula
The sum of terms of a finite geometric series is given by the following formula:
( n
a(r −1)
Sn = a(1−rr−1n , when r > 1
)
1−r , when r < 1
In this formula, Sn represents the sum of the first n terms of the geometric
series.
3.1.4 Sum of Infinite Geometric Series Formula
The sum of an infinite geometric series, which converges when −1 < r ̸= 0 < 1,
is given by:
a
S∞ = , −1 < r ̸= 0 < 1
1−r
This formula provides the sum of the entire infinite series.
3.2 Conclusion
Geometric series, or geometric progressions, are a fundamental concept in math-
ematics and are characterized by their common ratio. The formulas presented
here enable the calculation of specific terms, the sum of finite geometric series,
and the sum of infinite geometric series. These concepts have wide-ranging
applications in various fields of science and engineering.
