Sequence and Series | Mathematics for ACT PDF Download

Introduction

Sequence and series are used in mathematics as well as in our daily lives. A sequence is also known as progression and a series is developed by sequence. Sequence and series is one of the basic concepts in Arithmetic. Sequences are the grouped arrangement of numbers orderly and according to some specific rules, whereas a series is the sum of the elements in the sequence. For example, 2, 4, 6, 8 is a sequence with four elements and the corresponding series will be 2 + 4 + 6+ 8, where the sum of the series or value of the series will be 20.
There are various types of sequences and series depending upon the set of rules that are used to form the sequence and series. Sequence and series are explained in detail below.

What are Sequence and Series?

The sequence is the group or sequential arrangement of numbers in a particular order or set of rules. Series is formed by adding the terms of a sequence. In a sequence, an individual term can be present in many places. Sequences can be of two types, i.e. infinite sequence and finite sequence and series will be then defined by adding the terms of the sequence. Sum of infinite terms in a series is possible in some cases as well.
Let us understand this with an example. 1, 3, 5, 7, 9, 11, ... is a sequence where there is a common difference of 2 between any two terms and the sequence goes on increasing up to infinity unless the upper limit is given. These types of sequences are known as arithmetic sequences. Now if we add the numbers in the sequence like 1 + 3 + 5 + 7+ 9... this will make a series of this sequence. These kinds of series are known as arithmetic series. A few examples of sequence and series are given in the image shown below:

Sequence and Series

Arithmetic Sequence → 10, 15, 20, 25, 30, 35,..............., u_n
Arithmetic Series → 10 + 15 + 20 + 25 + 30 + 35 + ............... + u_n
Geometric Sequence → 100, 50, 25,12.5, 6.25, 3.125,..., u_n
Geometric Series → 100 + 50 + 25 + 12.5 + 6.25 + 3.125 + . . . + u_n

Difference Between Sequence and Series

The important differences between sequence and series are explained in the table given below:

Types of Sequence and Series

There are various types of sequences and series, in this section, we will discuss some special and most commonly used sequences and series. The types of sequence and series are:

• Arithmetic Sequence and Series
  An arithmetic sequence is a sequence where the successive terms are either the addition or subtraction of the common term known as common difference. For example, 1, 4, 7, 10, ...is an arithmetic sequence. A series formed by using an arithmetic sequence is known as the arithmetic series for example 1 + 4 + 7 + 10... is an arithmetic series.
• Geometric Sequence and Series 
  A geometric sequence is a sequence where the successive terms have a common ratio. For example, 1, 4, 16, 64, ...is an arithmetic sequence. A series formed by using geometric sequence is known as the geometric series for example 1 + 4 + 16 + 64... is a geometric series. The geometric progression can be of two types: Finite geometric progression and infinite geometric series.
• Harmonic Sequence and Series
  A harmonic sequence is a sequence where the sequence is formed by taking the reciprocal of each term of an arithmetic sequence. For example, 1, 1/4, 1/7, 1/10,... is a harmonic sequence. A series formed by using harmonic sequence is known as the harmonic series for example 1 + 1/4 + 1/7 + 1/10.... is a harmonic series.

Sequence and Series Formulas

There are various formulas related to various sequences and series by using them we can find a set of unknown values like the first term, nth term, common parameters, etc. These formulas are different for each kind of sequence and series. Formulas related to various sequences and series are explained below:

Arithmetic Sequence and Series Formula

The various formulas used in arithmetic sequence are given below:

Geometric Sequence and Series Formulas

The various formulas used in geometric sequence are given below:

Sequence and Series Tips

The following points are helpful to clearly understand the concepts of sequence and series.

• In an arithmetic sequence and series, a is represented as the first term, d is a common difference, a_n as the nth term, and n as the number of terms.
• In general, the arithmetic sequence can be represented as a, a + d, a + 2d, a + 3d,...
• Each successive term is obtained in a geometric progression by multiplying the common ratio to its preceding term.
• The formula for the nth term of a geometric progression whose first term is a and common ratio is r is a_n = arⁿ⁻¹
• The sum of the infinite GP formula is given as S_n = a/(1 − r) where |r|<1.

Sequence and Series Examples

Example 1: What will be the 15^th term of the arithmetic sequence -3, -(1/2), 2…. using sequence and series formula?
Solution: Given a = -3, d = -(1/2) -(-3) = 5/2, n = 15
Using the formula for n^th term of an arithmetic sequence: a_n = a+(n-1)d
Putting the known values:
a₁₅ = -3 +(15-1) 5/2
a₁₅ = 32

Example 2: Find the sum of the infinite geometric series -1 + 1/2 - 1/4 + 1/8 - 1/16 + ...
Solution: The common ratio of the given series is, r = -1/2.
Here, |r| = |-1/2| = 1/2 < 1.
Using the sequence and series formulas,
Sum of the given series = a / (1 - r)
= -1 / (1 - (-1/2))
= -1 / (3/2)
= -2/3