# Sequence and Series – Types, Difference, & Formulas

You have studied various patterns in mathematics such as shape patterns, letter patterns, number patterns, etc. Sequences and series are number patterns that are widely used in different fields. One of the many applications of sequences and series occurs in financial mathematics.

Let’s use examples to understand what a sequence and series are and the types and differences between the two.

## What are Sequence and Series?

A sequence is a group of numbers arranged in a particular order or following a set of rules. For example, $1$, $3$, $5$, $7$, … is a sequence that starts with a number $1$ and the rule to generate a next term is ‘add $2$ to the previous term’.

Therefore, we can say that a sequence is an arrangement of any objects or a set of numbers in a particular order followed by some rule. If $a_1$, $a_2$, $a_3$, $a_4$, … etc. denote the terms of a sequence, then $1$, $2$, $3$, $4$, … denotes the position of the term.

A sequence can be i.e. either a finite sequence or an infinite sequence depending on whether there are finite or finite terms in a sequence.

A series is formed by adding the terms of a sequence. For example $1 + 3 + 5 + 7 + …$ is a seeries. If $a_1$, $a_2$, $a_3$, $a_4$, … is a sequence, then the corresponding series is given by $\text{S}_n = a_1+ a_2 + a_3 + .. + a_n$.

## Sigma Notation of a Series

Series are often represented in compact form, called sigma notation, using the Greek letter $\sum{}$(sigma) as means of indicating the summation involved. Thus, the series $a_1 + a_2 + a_3 + … + a_n$ is abbreviated as $\sum_{k=1}^{n}{{a_k}}$.

Note: When the series is used, it refers to the indicated sum, not to the sum itself. For example, $1 + 3 + 5 + 7$ is a finite series with four terms. When we use the phrase “sum of a series,” we will mean the number that results from adding the terms, the sum of the series is $16$.

## General Term of a Sequence

The general term for a sequence follows a certain pattern. The successive terms are obtained by performing the mathematical operation present in the general term in indicated order to the previous term. Sometimes each term of the series follows an expression.

### Examples of General Term of a Sequence

Let’s consider some examples to understand the general term of a sequence.

Example 1: Write the first five terms of a sequence whose general term is $a_n = 3n – 5$.

The general term of the sequence is $a_n = 3n – 5$.

When $n = 1$, $a_1 = 3 \times 1 – 5 = 3 – 5 = -2$.

When $n = 2$, $a_2 = 3 \times 2 – 5 = 6 – 5 = 1$.

When $n = 3$, $a_3 = 3 \times 3 – 5 = 9 – 5 = 4$.

When $n = 4$, $a_4 = 3 \times 4 – 5 = 12 – 5 = 7$.

When $n = 5$, $a_5 = 3 \times 5 – 5 = 15 – 5 = 10$.

Therefore, the first five terms of the sequence are $-2$, $1$, $4$, $7$, and $10$.

Example 2: Write the first five terms of a sequence whose general term is $a_n = \frac{n + 2}{3}$.

The general term of the sequence is $a_n = \frac{n + 2}{3}$.

When $n = 1$, $a_1 = \frac{1 + 2}{3} = \frac{3}{3} = 1$.

When $n = 2$, $a_2 = \frac{2 + 2}{3} = \frac{4}{3}$.

When $n = 3$, $a_3 = \frac{3 + 2}{3} = \frac{5}{3}$.

When $n = 4$, $a_4 = \frac{4 + 2}{3} = \frac{6}{3} = 2$.

When $n = 5$, $a_2 = \frac{5 + 2}{3} = \frac{7}{3}$.

Therefore, the first five terms of the sequence are $1$, $\frac{4}{3}$, $\frac{5}{3}$, $2$, and $\frac{7}{3}$.

Example 3: Write the first five terms of a sequence whose general term is $a_n = (n + 1)(n – 2)(n + 3)$.

The general term of the sequence is $a_n = a_n = (n + 1)(n – 2)(n + 3)$.

When $n = 1$, $a_1 = (1 + 1)(1 – 2)(1 + 3) = 2 \times (-1) \times 4 = -8$.

When $n = 2$, $a_2 = (2 + 1)(2 – 2)(2 + 3) = 3 \times 0 \times 5 = 0$.

When $n = 3$, $a_3 = (3 + 1)(3 – 2)(3 + 3) = 4 \times 1 \times 6 = 24$.

When $n = 4$, $a_4 = (4 + 1)(4 – 2)(4 + 3) = 5 \times 2 \times 7 = 70$.

When $n = 5$, $a_5 = (5 + 1)(5 – 2)(5 + 3) = 6 \times 3 \times 8 = 144$.

Therefore, the first five terms of the sequence are $-8$, $0$, $24$, $70$, and $144$.

Example 4: Write the first five terms of a sequence which is defined as $a_1 = 1$, $a_n = a_{n-1} + 2, n \ge 2$.

Here the first term is $a_1 = 1$ and the next term $a_n$ is given in terms of the previous term $a_{n – 1}$, defied by $a_n = a_{n-1} + 2$.

When $n = 1$, $a_1 = 1$.

When $n = 2$, $a_2 = a_{2-1} + 2 = a_1 + 2 = 1 + 2 = 3$.

When $n = 3$, $a_3 = a_{3-1} + 2 = a_2 + 2 = 3 + 2 = 5$.

When $n = 4$, $a_4 = a_{4-1} + 2 = a_3 + 2 = 5 + 2 = 7$.

When $n = 5$, $a_5 = a_{5-1} + 2 = a_4 + 2 = 7 + 2 = 9$.

Therefore, the first five terms of the sequence are $1$, $3$, $5$, $7$, and $9$.

Example 5: Write the first six terms of a sequence which is defined as $a_1 = a_2 = 2$, $a_n = a_{n – 1} – 1, n > 2$.

Here the first two terms is $a_1 = 2$, and $a_2 = 2$ and the next term $a_n$ is given in terms of the previous term $a_{n – 1}$, defied by $a_n = a_{n – 1} – 1$.

When $n = 1$, $a_1 = 1$.

When $n = 2$, $a_2 = 1$.

When $n = 3$, $a_3 = a_{3 – 1} – 1 = a_2 – 1 = 1 – 1 = 0$.

When $n = 4$, $a_4 = a_{4 – 1} – 1 = a_3 – 1 = 0 – 1 = -1$.

When $n = 5$, $a_5 = a_{5 – 1} – 1 = a_4 – 1 = -1 – 1 = -2$.

When $n = 6$, $a_6 = a_{6 – 1} – 1 = a_5 – 1 = -2 – 1 = -3$.

Therefore, the first six terms of the sequence are $1$, $1$, $0$, $-1$, $-2$, and $-3$.

Example 6: Write the first six terms of a sequence which is defined as $a_1 = a_2 = 1$ and $a_n = a_{n – 1} + a_{n – 2}, n > 2$.

Here the first two terms is $a_1 = 1$, and $a_2 = 1$ and the next term $a_n$ is given in terms of the previous terms $a_{n – 1}$ and $a_{n – 2}$, defied by $a_n = a_{n – 1} + a_{n – 2}$.

When $n = 1$, $a_1 = 1$.

When $n = 2$, $a_2 = 1$.

When $n = 3$, $a_3 = a_{3 – 1} + a_{3 – 2} = a_2 + a_1 = 1 + 1 = 2$.

When $n = 4$, $a_4 = a_{4 – 1} + a_{4 – 2} = a_3 + a_2 = 2 + 1 = 3$.

When $n = 5$, $a_5 = a_{5 – 1} + a_{5 – 2} = a_4 + a_3 = 3 + 2 = 5$.

When $n = 6$, $a_6 = a_{6 – 1} + a_{6 – 2} = a_5 + a_4 = 5 + 3 = 8$.

Therefore, the first six terms of the sequence are $1$, $1$, $2$, $3$, $5$, and $8$.

Note: This sequence is a very popular sequence and is known as Fibonacci Sequence.

## Difference Between Sequence and Series

The following are the differences between sequence and series.

## Types of Sequences and Series

There are various types of sequences and series. The most common of these are

• Arithmetic Sequences 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, $5$, $9$, $13$, $17$, $21$, … is an arithmetic sequence. Here the first term $a = 5$, and the common difference $d = 9 – 5 = 13 – 9 = 17 – 13 = 4$. A series formed by using an arithmetic sequence is known as the arithmetic series for example $5 + 9 + 13 + 17…$ is an arithmetic series.
• Geometric Sequences and Series: A geometric sequence is a sequence where the successive terms have a common ratio. For example, $1$, $4$, $16$, $64$, … is a geometric sequence. Here the first term $a = 1$, and the common ratio $r = \frac{4}{1} = \frac{16}{4} = \frac{64}{16} = 4$. A series formed by using a geometric sequence is known as a geometric series for example $1 + 4 + 16 + 64 …$ is a geometric series.
• Harmonic Sequences 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, $\frac{1}{5}$, $\frac{1}{9}$, $\frac{1}{13}$,$\frac{1}{17}$, $\frac{1}{21}$, … is a harmonic sequence. A series formed by using harmonic sequence is known as a harmonic series for example $\frac{1}{5} + \frac{1}{9} + \frac{1}{13} + \frac{1}{17}+ \frac{1}{21} …$ is a harmonic series.

## Key Takeaways

• In an arithmetic sequence and series, $a$ is represented as the first term, $d$ is a common difference, $a_n$ is the $n^{th}$ term, and $n$ is the number of terms.
• In general, the arithmetic sequence can be represented as $a$, $a+d$, $a+2d$, $a+3d$, .., where $a$ is the first term, $d$ is the common difference.
• Each successive term is obtained in a geometric progression by multiplying the common ratio by its preceding term.
• The formula for the nth term of a geometric progression whose first term is $a$ and the common ratio is $r$ is $a_n = ar_{n−1}$
• The sum of the infinite GP formula is given as $\text{S}_n = \frac{a}{1−r}$ where $|r|<1$.

## Practice Problems

1. What is a sequence?
2. What is a series?
3. What is the difference between sequence and series?
4. Write the first six terms of a sequence which is defined as
• $a_n = n(n + 3)$
• $a_n = \frac{2n}{n – 1}$
• $a_n = 2^{n – 1}$
• $a_n = 2.3^n$
• $a_n = \frac{4n – 5}{6}$
• $a_n = (-1)^n.4^{-n}$
• $a_n = (-1)^{n-1}.2^n$
• $a_n = \frac{n \left(n^2 – 4 \right)}{3}$
• $a_1 = -2$, $a_n = 3a_{n-1} + 2$
• $a_1 = 1$, $a_n = \frac{a_{n-1}}{2}$

## FAQs

### Give an example of sequence and series.

An example of a sequence is $1$, $4$, $7$, $10$, $13$, … An example of a series is $1 + 4 + 7 + 10 + 13 + …$

### What are some of the common types of sequences?

The three most common types of sequences are arithmetic sequence, geometric sequence, and harmonic sequence.

### What is the difference between sequence and series?

In sequence, elements are placed in a particular order following a particular set of rules, a definite pattern of the numbers is important, and the order of appearance of the numbers is important. In series, the order of the elements is not necessary, the pattern of the numbers is not important, and the order of appearance is not important.

### What is the similarity between sequence and series?

The sequence and the series of the same type, both are made up of the same elements, i.e, the elements that follow a pattern. A series is formed by using the elements of the sequence and adding them by the addition symbol.

### What sigma notation of a series?

Series are often represented in compact form, called sigma notation, using the Greek letter $\sum{}$(sigma) as means of indicating the summation involved. Thus, the series $a_1 + a_2 + a_3 + … + a_n$ is abbreviated as $\sum_{k=1}^{n}{{a_k}}$.

## Conclusion

A sequence is a group of numbers arranged in a particular order or following a set of rules, whereas a series is formed by adding the terms of a sequence. If $a_1$, $a_2$, $a_3$, $a_4$, … etc. denote the terms of a sequence, then the corresponding series is given by $\text{S}_n = a_1+ a_2 + a_3 + .. + a_n$.