Sequences and Series: Definition, Formulas & Examples

Sequences and Series: A sequence is a collection of objects (or events) arranged logically in mathematics. A sum of a sequence of terms is referred to as a series. In other words, a series is a collection of numbers connected by addition operations. In this article, we shall discuss about sequence, the different standard series like the arithmetic, geometric and harmonic series etc.

What are Sequence and Series?

Sequence: A sequence is a group of numbers called terms that are arranged in a specific order. The difference between two consecutive terms in an arithmetic series is always the same. The distinction is known as the common difference. A geometric sequence is one in which the ratio of two successive terms remains constant.

Series: The series of a sequence is the sum of the sequence to a certain number of terms. It is often written as \(S_{n}\). So, if the sequence is \(2,4,6,8,10, \ldots\)

The sum to \(3\) terms \(=S_{3}=2+4+6=12\)

Types of Sequence and Series

There are many different kinds of sequences and series. In this section, we will go through a few of the most frequent ones. Sequences and series can be divided into the following categories:

1. Arithmetic sequences and series
2. Geometric sequences and series
3. Harmonic sequences and series

Examples of Sequence and Series

Some of the examples of the sequences and series are:

Arithmetic Sequence: An arithmetic sequence is one in which each term is formed by adding or subtracting a defined number from the previous number.

1. \(1,2,3,4, \ldots\)
2. \(100,70,40,10, \ldots\)
3. \(-3,-2,-1,0, \ldots\)

Geometric Sequence: A geometric sequence is one in which each term is obtained by multiplying or dividing a defined integer by the previous number.

1. \(2,4,8,16,32, \ldots\)
2. \(1,-1,1,-1,1, \ldots\)
3. \(2, \quad \frac{2}{3}, \quad \frac{2}{9}, \quad \frac{2}{27}, \frac{2}{81}, \ldots\)

Harmonic Sequence: If the reciprocals of all the elements in a sequence are taken from an arithmetic sequence, then it is said to be in a harmonic sequence.

1. \(\frac{1}{5}, \frac{1}{10}, \frac{1}{15}, \frac{1}{20}, \ldots\)
2. \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots\)
3. \(\frac{1}{4}, \frac{1}{7}, \frac{1}{10}, \frac{1}{13}, \ldots\)

Sequence and Series Formulas

The formulas for the arithmetic, geometric, and harmonic series can be found below. The formulas for finding the \(n^{\text {th }}\) term and the sum of the \(n\) terms of the series are included in the sequence and series formulas. There is a common difference between two successive terms in an arithmetic series and a common ratio between consecutive terms in a geometric series.

Formula for nth Term of an Arithmetic Series

An arithmetic series \(n^{t h}\) term is written as follows.

The first term is represented by \(a\), while the common difference between two subsequent terms is represented by \(d\). Each subsequent term has one more \(d\) than the previous one. In the \({n^{{\rm{th}}}}\) term of the series, there are \((n-1) d^{\prime} s\).

Arithmetic series is given by,

\(a, a+d, a+2 d, a+3 d, a+4 d, \ldots a+(n-1) d\)

\(a_{1}=a, a_{2}=a+d, a_{3}=a+2 d, a_{n}=a+(n-1) d\)

Formula for Sum of n Terms of an Arithmetic Series

The sum of \(n\) terms of an arithmetic series is as follows:

The product of \(\frac{n}{2}\) and the sum of the first term \((a)\) and the last term \((a+(n-1) d)\) of the arithmetic series equals the sum of the \(n\) terms.

\(S_{n}=\frac{n}{2} \times(\text {First term} + \text {Last term})\)

\(S_{n}=\frac{n}{2} \times(2 a+(n-1) d)\)

Formula for nth Term of a Geometric Series

The \(n^{\text {th }}\) term of a geometric series is as follows.

The first term is \(a\), and the common ratio is \(r\). Each subsequent term of the geometric series is one more than the power of \(r\) of the previous term.

Geometric series is given by,

\(a, a r, a r^{2}, a r^{3}, a r^{4}, \ldots a r^{n-1}\)

\(a_{1}=a, a_{2}=a r, a_{3}=a r^{2}\)

\(n^{\text {th }}\) term of the geometric series is \(a_{n}=a r^{n-1}\)

Formula for Sum of n Terms of a Geometric Series

The sum of \(n\) terms and the sum of infinite terms of a geometric series are given below:

Sum of \(n\) terms of a geometric series \(S_{n}=a \frac{\left(r^{n}-1\right)}{(r-1)}\), for \(r>1\), and \(S_{n}=a \frac{\left(1-r^{n}\right)}{(1-r)}\), for \(r<1\).

Sum of infinite terms of a geometric series \(S_{n}=\frac{a}{1-r}\)

Formula for nth Term of Harmonic Series

The \(n^{th}\) term of a harmonic series is as follows.

The reciprocal of the arithmetic series is the harmonic series.

Hamonic series is given by \(\frac{1}{a}, \frac{1}{(a+d)}, \frac{1}{(a+2 d)}, \frac{1}{(a+3 d)}, \ldots\)

\(n^{\text {th }}\) term of the Harmonic series is given by,

\(a_{n}=\frac{1}{a+(n-1) d}\)

Difference Between Sequence and Series

Uses of Sequence and Series

1. Sequences and series play a significant role in our lives in a variety of ways. They aid in decision-making by assisting us in predicting, evaluating, and monitoring the outcome of a situation or occurrence.
2. In business and financial analysis, mathematical sequences and series are used to aid decision-making and discover the optimum solution to a problem.
3. The convergence features of sequences are important in a variety of mathematical disciplines for understanding functions, spaces, and other mathematical structures.
4. Sequences, in particular, form the foundation for series, which are crucial in differential equations and analysis.
5. Sequences and series are fundamental in mathematics and have a wide range of applications in finance, physics, and statistics.

Solved Examples – Sequences and Series

Q.1. Find the value of the \(24^{\text {th }}\) term of the arithmetic series \(5,9,13,17 \ldots\).
Ans: The given arithmetic series is \(5,9,13,17 \ldots..\)
First-term in the series \(a=5\) Common difference in the \(d=9-5=4\)
The \(24^{\text {th }}\) term \(=T_{24}=a+23 d\)
\(=5+23 \times 4=5+92=97\)
Therefore, \(24^{\text {th }}\) term of the series is \(97\).

Q.2. The fifth term of the arithmetic series is \(23\), and the seventh term of the series is \(31\). Find the first four terms of the series.
Ans: From the given, \(T_{5}=23, T_{7}=31\)
\(a+6 d=31 \ldots..(i)\)
\(a+4 d=23 \ldots..(ii)\)
The difference of the equation \((i)\) and \((ii)\) is
\(2 d=8\)
\(d=\frac{8}{2}=4\)
Then, \(a=7\)
So, \(a=7, a+d=11, a+2 d=15, a+3 d=19\)
Therefore, the first four terms of the series \(7,11,15\) and \(19\).

Q.3: Find the sum of geometric series \(8+4+2+1+\ldots\).
Ans: Given geometric series is \(8+4+2+1+\ldots\).
From the series \(a=8, r=\frac{1}{2}\)
The formula to find the sum of the geometric series is:
\(S_{n}=\frac{a}{1-r}\)
So, here, \(S_{n}=\frac{8}{1-\frac{1}{2}}=\frac{8}{{\frac{{2 – 1}}{2}}}=\frac{16}{1}=16\)
Therefore, the sum of the given geometric series is \(16\).

Q.4. Find the sum of the terms of the series \(5+9+13+\ldots+20\).
Ans: The given is an arithmetic series with \(a=5, d=4\).
\(S_{n}=\frac{n}{2} \times(2 a+(n-1) d)\)
\(S_{20}=\frac{20}{2} \times(2(5)+(20-1) 4)\)
\(S_{20}=10 \times(10+76)\)
\(S_{20}=10 \times 86\)
\(S_{20}=860\)
Therefore, the obtained sum is \(860\).

Q.5. The sixth term of an H.P is \(10\), and the \(11^{\text {th }}\) term is \(18\). Find the \(16^{\text {th }}\) term.
Ans: From the given, \(a+5 d=\frac{1}{10}, a+10 d=\frac{1}{18}, a+15 d=?\)
\(a+5 d=\frac{1}{10} \ldots \ldots..(i)\)
\(a+10 d=\frac{1}{18} \ldots..(ii)\)
Solving the equation \((i)\) and \((ii)\) we get
\(d=\frac{-2}{225}, a=\frac{13}{90}\)
So, \(a+15 d=\frac{13}{90}+15\left(\frac{-2}{225}\right)\)
\(\Rightarrow a+15 d=\frac{13}{90}-\frac{2}{15}\)
\(\Rightarrow a+15 d=\frac{13-12}{90}\)
\(\Rightarrow a+15 d=\frac{1}{90}\)
Therefore, the obtained \(16^{\text {th }}\) term is \(90\).

Summary

A sequence is a logical arrangement of items (or events), with each member occurring before or after the others. A series is defined as the sum of a set of terms. In other words, a series is a group of numbers linked together by addition operations. This article includes the definitions of sequence and series, types, formulas, differences, and uses of sequence and series.

This article helps in better understanding the topic sequence and series. The outcome of this article helps in apply the suitable formulas while solving the various problems based on them.

Learn All the Concepts on Arithmetic Progression

Frequently Asked Questions (FAQs) – Sequences and Series

Q.1. What are sequences and series?
Ans: A sequence is a collection of objects (or events) arranged logically. 
A sum of a sequence of terms is referred to as a series. In other words, a series is a collection of numbers connected by addition operations.

Q.2. What is the importance of sequence and series?
Ans: The importance of sequence and series are listed below,
1. Sequences and series play a significant role in our lives in a variety of ways. They aid in decision-making by assisting us in predicting, evaluating, and monitoring the outcome of a situation or occurrence.
2. In business and financial analysis, mathematical sequences and series aid decision-making and discover the optimum solution to a problem.

Q.3. What is the formula for sequence and series?
Ans: Formula to find the \(n^{\text {th }}\) term of the arithmetic series is given by,
\(a_{n}=a+(n-1) d\)
Formula to find the sum terms of the arithmetic series is given by,
\(S_{n}=\frac{n}{2} \times(2 a+(n-1) d)\)

Q.4. What are the differences between sequence and series?
Ans: Some of the differences between a sequence and series are listed below:

Q.5: What are the types of sequence?
Ans: Types of sequences are given by:
1. Arithmetic sequences
2. Geometric sequences
3. Harmonic sequences

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