# Sequences and Series Formulas

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Sequence and series formulas are related to different types of sequences and series in math. A sequence is the set of ordered elements that follow a pattern and a series is the sum of the elements of a sequence. The sequence and series formulas generally include the formulas for the nth term and the sum.

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## Sequence And Series

Sequence and series are the basic topics in Arithmetic. An itemized collection of elements in which repetitions of any sort are allowed is known as a sequence, whereas a series is the sum of all elements. An arithmetic progression is one of the common examples of sequence and series.

• In short, a sequence is a list of items/objects which have been arranged in a sequential way.
• A series can be highly generalized as the sum of all the terms in a sequence. However, there has to be a definite relationship between all the terms of the sequence.

The fundamentals could be better understood by solving problems based on the formulas. They are very similar to sets but the primary difference is that in a sequence, individual terms can occur repeatedly in various positions. The length of a sequence is equal to the number of terms and it can be either finite or infinite.  This concept is explained in a detailed manner in Class 11 Maths. With the help of definition, formulas and examples we are going to discuss here the concepts of sequence as well as series.

## Sequence and Series Definition

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,…..denotes the position of the term.

A sequence can be defined based on the number of terms i.e. either finite sequence or infinite sequence.

If a1, a2, a3, a4, ……. is a sequence, then the corresponding series is given by

SN = a1+a2+a3 + .. + aN

Note:  The series is finite or infinite depending if the sequence is finite or infinite.

## Types of Sequence and Series

Some of the most common examples of sequences are:

• Arithmetic Sequences
• Geometric Sequences
• Harmonic Sequences
• Fibonacci Numbers

Sequences: A finite sequence stops at the end of the list of numbers like a1, a2, a3, a4, a5, a6……an. whereas, an infinite sequence is never-ending i.e. a1, a2, a3, a4, a5, a6……an…..

Series: In a finite series, a finite number of terms are written like a+ a+ a3 + a+ a5 + a6 + ……an. In case of an infinite series, the number of elements are not finite i.e. a+ a+ a3 + a+ a5 + a6 + ……a+…..

### Arithmetic Sequences

A sequence in which every term is created by adding or subtracting a definite number to the preceding number is an arithmetic sequence.

Example:

3,  6,  9,  12, 15,  18, 21……..

+3 +3  +3  +3  +3  +3

Here, the difference between the two successive terms is 3. So, it is called the difference.

When we have to get the next number of this sequence, we simply add 3 to the last number of this sequence.

The difference is represented by “d”.

In the above example, we can see that a1 =3 and a2 = 6.

The difference between the two successive terms is

a2 – a1 = 3

a3 – a2 = 3

In an arithmetic sequence, if the first term is a1 and the common difference is d, then the nth term of the sequence is given by:

an=a1+(n−1)d

Using the above formula, we can successfully determine any number of any given arithmetic sequence.

### Geometric Sequences

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.

### Harmonic Sequences

A series of numbers is said to be in harmonic sequence if the reciprocals of all the elements of the sequence form an arithmetic sequence.

Example:

1, 4, 16, 64…

Here,

a1 =1

a2 = 4 = a1(4)

a3 = 16 = a2(4)

Here, we are multiplying it by 4 every time to get the next term. The ratio here is 4.

The ratio is denoted by “r”.

an = an-1⋅r or

an = a1⋅rn−1

an=an−1×r

Using the above formula, we can determine any number of any given geometric sequence.

### Fibonacci Numbers

Fibonacci numbers form an interesting sequence of numbers in which each element is obtained by adding two preceding elements and the sequence starts with 0 and 1. Sequence is defined as, F0 = 0 and F1 = 1 and Fn = Fn-1 + Fn-2

Example:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 54, 88, 142  …

In the above sequence, we can see

a1 =0, a2 = 1

a3 = a2 + a1 = 0 + 1 =1

a4 = a3 + a2  = 1 + 1 =2, and so on.

So, the Fibonacci Sequence formula is

an = an-2 + an-1, n > 2

This is also called the Recursive Formula. Using this formula, we can calculate any number of the Fibonacci sequence.

## Sequence and Series Formulas

List of some basic formula of arithmetic progression and geometric progression are

*Here, a = first term, d = common difference, r = common ratio, n = position of term, l = last term

## Difference Between Sequences and Series

Let us find out how a sequence can be differentiated with series.

## Sequence and Series Examples

Question 1: If 4,7,10,13,16,19,22……is a sequence, Find:

1. Common difference
2. nth term
3. 21st term

Solution: Given sequence is, 4,7,10,13,16,19,22……

a) The common difference = 7 – 4 = 3

b) The nth term of the arithmetic sequence is denoted by the term Tn and is given by Tn = a + (n-1)d, where “a” is the first term and d is the common difference.
Tn = 4 + (n – 1)3 = 4 + 3n – 3 = 3n + 1
c) 21st term as:  T21 = 4 + (21-1)3 = 4+60 = 64.

Question 2: Consider the sequence 1, 4, 16, 64, 256, 1024….. Find the common ratio and 9th term.

Solution: The common ratio (r)  = 4/1 = 4

The preceding term is multiplied by 4 to obtain the next term.

The nth term of the geometric sequence is denoted by the term Tn and is given by Tn = ar(n-1)
where a is the first term and r is the common ratio.

Here a = 1, r = 4 and n = 9

So, 9th term is can be calculated as T9 = 1* (4)(9-1)= 48 = 65536.

### What does a Sequence and a Series Mean?

A sequence is defined as an arrangement of numbers in a particular order. On the other hand, a series is defined as the sum of the elements of a sequence.

### What are Some of the Common Types of Sequences?

A few popular sequences in maths are:

• Arithmetic Sequences
• Geometric Sequences
• Harmonic Sequences
• Fibonacci Numbers

### What are Finite and Infinite Sequences and Series?

Sequences: A finite sequence is a sequence that contains the last term such as a1, a2, a3, a4, a5, a6……an. On the other hand, an infinite sequence is never-ending i.e. a1, a2, a3, a4, a5, a6……an…..

Series: In a finite series, a finite number of terms are written like a1 + a2 + a3 + a4 + a5 + a6 + ……an. In case of an infinite series, the number of elements are not finite i.e. a1 + a2 + a3 + a4 + a5 + a6 + ……an +…..

### Give an example of sequence and series.

An example of sequence: 2, 4, 6, 8, …
An example of a series: 2 + 4 + 6 + 8 + …

### What is the formula to find the common difference in an arithmetic sequence?

The formula to determine the common difference in an arithmetic sequence is:
Common difference = Successive term – Preceding term.

### How to represent the arithmetic sequence?

If “a” is the first term and “d” is the common difference of an arithmetic sequence, then it is represented by a, a+d, a+2d, a+3d, …

### How to represent the geometric sequence?

If “a” is the first term and “r” is the common ratio of a geometric sequence, then the geometric sequence is represented by a, ar, ar2, ar3, …., arn-1, ..

### How to represent arithmetic and geometric series?

The arithmetic series is represented by a + (a+d) + (a+2d) + (a+3d) + …
The geometric series is represented by a + ar + ar2 + ar3 + ….+ arn-1+ ..

## What are Sequences and Series Formulas?

The below list includes sequences and series formulas for the arithmetic, geometric, and harmonic sequences. Here the sequence and series formulas include formulas

• to find the nth term of the sequence and
• to find the sum of the n terms of the series.

In an arithmetic sequence, there is a common difference between two subsequent terms. In a geometric sequence, there is a common ratio between consecutive terms. In a harmonic sequence, the reciprocals of its terms are in an arithmetic sequence. The figure below shows all sequences and series formulas.

Let us see each of these formulas in detail and understand what each variable represents.

### Arithmetic Sequence and Series Formulas

Consider the arithmetic sequence a, a+d, a+2d, a+3d, a+4d, …., where ‘a’ is its first term and ‘d’ is its common difference. Then:

• nth term of arithmetic sequence, an = a + (n – 1) d
• Sum of the arithmetic series, Sn = n/2 (2a + (n – 1) d) (or) Sn = n/2 (a + an)

### Geometric Sequence and Series Formulas

Consider the geometric sequence a, ar, ar2, ar3, …, where ‘a’ is the first term and ‘r’ is the common ratio. Then:

• nth term of geometric sequence, an = a rn – 1
• Sum of the finite geometric series (sum of first ‘n’ terms), Sn = a (1 – rn) / (1- r)
• Sum of infinite geometric series, Sn = a / (1 – r) when |r| < 1. The sum is not defined when |r| ≥ 1.

### Harmonic Sequence and Series Formulas

Consider the harmonic sequence 1/a, 1/(a+d), 1/(a+2d), 1/(a+3d), 1/(a+4d), …., where ‘1/a’ is its first term and ‘d’ is the common difference of the arithmetic sequence a, a + d, a + 2d, …. Then:

• nth term of harmonic sequence, an = 1 / (a + (n – 1) d)
• Sum of the harmonic series, Sn = 1/d ln [ (2a + (2n – 1) d) / (2a – d) ]

## Examples on Sequences and Series Formulas

Example 1: Find the value of the 25th term of the arithmetic sequence 5, 9, 13, 17…..

Solution:

The given sequence is 5, 9, 13, 17…..

The first term, a = 5

The common difference, d = 9 – 5 = 4

Using the sequence and series formulas,

an = a + (n – 1) d

For the 25th term, substitute n = 25:

a25 = a + 24d = 5 + 24*4 = 5 + 96 = 101

Answer: Hence the 25th term of the series is 101.

Example 2: Find the sum of the first 100 terms of the arithmetic series 1 + 4 + 7 + ….

Solution:

In the given series, the first term is a = 1 and the common difference is d = 3.

Using the sequences and series formulas,

Sn = n/2 (2a + (n – 1) d)

For the sum of 100 terms, substitute n = 100:

S100 = 100/2 (2(1) + (100 – 1) 3) = 14,950

Answer: The sum of the first 100 terms is 14,950.

Example 3: Is it possible to find the sum of the infinite geometric series 5, -5/2, 5/4, -5/8, …? If so, find the sum.

Solution:

In the given geometric series, the common ratio, r = -1/2. So |r| = |-1/2| = 1/2 < 1. So it is possible to find its sum using one of the sequence and series formulas:

S = a / (1 – r)

= 5 / (1 – (-1/2))

= 5 / (3/2)

= 10/3

Answer: Sum of all terms of the given series = 10/3.

## FAQs on Sequences and Series Formulas

### List some Important Sequences and Series Formulas.

The sequences and series formulas for different types are tabulated below:

### What is the Difference Between Sequence and Series Formulas?

The sequence formulas would tell how to find the nth term (or general term) of a sequence whereas the series formulas would tell us how to find the sum (series) of a sequence.

### When to Use Sequences and Series Formulas?

To find the nth term of a specific sequence, use the sequence formulas. To find the sum of terms of a sequence, use the series formulas.

### What is the Sum of a Harmonic Series Using Sequences and Series Formulas?

For a harmonic sequence 1/a, 1/(a+d), 1/(a+2d), 1/(a+3d), 1/(a+4d), …., the sum of its first ‘n’ terms can be found using the formula Sn = 1/d ln [ (2a + (2n – 1) d] / (2a – d) ].

## Solved Examples

#### Question 1: If 1, 3, 5, 7, 9…… is a sequence, Find Common difference, nth term, 21st term

Solution: Given sequence is, 1, 3, 5, 7, 9……

1. a) common difference d = 3 – 1 = 2
2. b) The nth term of the arithmetic sequence is denoted by the term Tnand is given by Tn = a + (n-1) d,
3. c) 21st term as:  T21= 1 + (21-1)2 = 1+40 = 41.

### Solved Examples

Example 1: What will be the 6th number of the sequence if the 5th term is 12 and the 7th term is 24?

Solution: As the two numbers are given, the 6th number will be the arithmetic mean of the two given numbers.

AM = 12 + 24 / 2

= 36/2

= 18

Hence, the 6th term will be 18.

Example 2: Find the geometric mean of 2 and 18.

Solution: Formula to calculate the geometric mean.

p = 2 and q = 18

### Quiz Time

1. What is the ninth term of the geometric sequence, 3, 6, 12, 24, …?

Sol: Given the first term is 3.

Common ration=63=2

So, ninth term a9=ar8=328

=3356

=768.

3. What is the sum of the first ten terms of the geometric sequence 5, 15, 45, …?

Sol: From a given series, we can determine that the first term is 5.

Common ratio=153=5

Formula to find sum of n terms = a1(rn-1)r-1

=5(310-1)3-1

=5590482

=147620

## FAQs on Sequence and Series Formula

1. Difference Between Sequence and Series

There is a lot of confusion between sequence and series, but you can easily differentiate between them as follows:

• A sequence is a particular format of elements in some definite order, whereas series is the sum of the elements of the sequence.
• In sequence, order of the elements is definite, but in series, the order of elements is not fixed.
• A sequence is represented as 1,2,3,4,….n, whereas the series is represented as 1+2+3+4+…..n.
• In sequence, the order of elements has to be maintained, whereas in series, the order of elements is not important.

2. What is an Arithmetic Series?

An arithmetic series is what you get when you add up all the terms of a sequence. The resulting values are called the “sum” or the “summation”. Series is indicated by either the Latin capital letter “S” or the Greek letter corresponding to the capital “S”, which is called “sigma” (SIGG-mug): written as Σ.

To show the summation of tenth terms of a sequence {an}, we would write as

Where “n = 1” is called the “lower index”, which represents that the series starts from 1 and the “upper limit” is 10; it means that the last term will be 10. And “an” stands for the terms that we’ll be adding. It is read as “the sum, from n equals one to ten, of a-sub-n”.

### Sample Problems

Problem 1: Using the sequence and series formula, determine the seventh term of the given geometric sequence: 3, 1, 1/3, 1/9, 1/27, 1/81, ___.

Solution:

Given sequence: 3, 1, 1/3, 1/9, 1/27, 1/81, ___

Now, a = 3, r = 1/3

By using the formula for the nth term of a geometric sequence and series:

an = ar(n-1)

Putting the known values in the formula:

a7 = 3 × (1/3)(7-1)

a7 = 3 × (1/3)6

a7 = (1/3)5 = 1/243

Hence, the seventh term of the given series is 1/243.

Problem 2: Using the sequence and series formula, find the 10th term of the arithmetic sequence 14, 10, 6, 2, -2, -6, ___.

Solution:

Given sequence: 14, 10, 6, 2, -2, -6, ___

Now, a = 14

d = 10 -14 = -4

Using the formula for the nth term of an arithmetic sequence:

an = a+(n-1)d

a10 = 14 + (10 – 1)(-4)

a10 = 14 + (9)(-4)

a10 = 14 – 36 = -22

Hence, the 10th term of the sequence is -22.

Problem 3: If p, q, and r are in A.P., find the value of (q2-pr)/(p – q)2.

Solution:

Given that p, q, and are in A.P

let p, q, and r be a-d, a, a + d.

So, p = a-d, q = a, r = a + d

p – q = a- d – (a + d) = -2d

(p – q)2 = (-2d)2 = 4d2

q2 = a2

p × r = (a – d) (a + d) = (a2 – d2)

q2 – pr = a2 – (a2 – d2) = d2

So, (q2 – pr)/(p – q)2 = d2/4d2 = 1/4

Hence, the value (q2-pr)/(p – q)2 = 1/4.

Problem 4: Find the sum of the infinite geometric series 1, -2/3,  4/9, -8/27, 16/81___.

Solution:

Given sequence: 1, – 2/3,  4/9, -8/27, 16/81___

Now, a = 1,

The common ration of the sequence, r = (-2/3)/1 = -2/3

By using the sequence and series formulas,

Sum of the given series = a/(1 – r)

= 1/(1 – (-2/3))

= 1/(1 + 2/3)

= 1/(5/3) = 3/5

Hence, the sum of the infinite geometric series is 3/5.

Problem 5: Determine the sum of the first 15 terms of the sequence 0.5, 0.55, 0.555,___ up to 15 terms.

Solution:

Given sequence: 0.5, 0.55, 0.555,___up to 15 terms

⇒ 0.5 + 0.55 + 0.555 + 0.5555, …….. up to 15 terms

⇒ 5[0.1 + 0.11 + 0.111 + 0.1111, …….. up to 15 terms]

⇒ (5/9)[0.9 + 0.99 + 0.999 + 0.9999, …… up to 15 terms]

⇒ (5/9) [(1 – 0.1) + (1 – 0.01) + (1 – 0.001), …… up to 15 terms]

⇒ (5/9) [(1 + 1 + 1 + 1, ……. up to 15 terms) – (0.1 + 0.01 + 0.001 + 0.0001 + ….. up o 15 terms)]

⇒ (5/9) [15 – (0.1) (1 – (0.1)15)/(1 – 0.1)]

⇒ (5/9) [15 – (0.1)(1 – (0.1)15)/(0.9)]

⇒ (5/9) [15 – (1/9) {1 – (0.1)15}]            as 1 – (0.1)15 = 1 (approx)

⇒ (5/9) (1/9) [134 ]

⇒ 8.27 (approx)

Problem 6: Determine the nth term of the given series: 2, (2 + 4), (2 + 4 + 6), (2 + 4 + 6 + 8),…..

Solution:

Here by observing the sequence,

nth term = (2 + 4 + 6 + 8 + 10 . . . . . . . . . . . .+ 2n)

The nth term is an arithmetic series in itself with first term (a) = 2 and common difference (d) = 2

Now,

Sum of n terms of an Arithmetic progression is (n/2)[2a + (n-1)d]

= (n/2)[2 × 2 + (n-1) × 2]

= (n/2) × 2 [ 2 + (n – 1)]

= n(n+1)

Hence, the nth term of the given series is n(n+1).