In mathematics, a series has a constant difference between terms. We can find out the sum of the terms in arithmetic series by multiplying the number of times the average of the last and first terms. Thus we can see that series and finding the sum of the terms of series is a very important task in mathematics. In this topic, we will discuss some popular series and series formula with examples. Let us learn the concept!
What is the Series?
What exactly is a series? Actually, a series in math is simply the sum of the various numbers or elements of the sequence. For example, to make a series from the sequence of the first five positive integers 1, 2, 3, 4, 5 we will simply add them up. Therefore 1 + 2 + 3 + 4 + 5 is a series.
So, series of a sequence is the sum of the sequence to some given number of terms, or sometimes till infinity. It is often written as S_n.
If the sequence is 2, 4, 6, 8, 10, … , then the sum of the first 3 terms:
S3=2+4+6
S3=12.
The Greek capital sigma i.e. Σ is usually used to represent the sum of a sequence. This may be best explained using the example:
This means to replace the term ‘i’ in the expression by 1 and write down what we get. Then replace ‘i’ by 2 and write down what we get. Keep doing this until we get to 4 since this is the number above the Sum.
Some Popular Series
1] Arithmetic Progressions
An arithmetic progression is a sequence, in which each term is a certain number larger than the previous term. The terms in the sequence are known as to increase by some common difference, d.
In general, the nth term of the arithmetic progression, given the first term ‘a’ and common difference ‘d’ will be an=a+(n–1)d
2]Geometric Progressions
A geometric progression is a sequence in which each term is r times larger than the previous term. Here r is called the common ratio of the sequence. Then the nth term of a geometric progression is as given :
an=arn−1
The first term is denoted as ‘a’ and the common ratio is denoted as ‘r’
3]Harmonic Progressions
Where l is last term.
2] The sum of a geometric progression
The sum of the n terms of a geometric progression is given as:
3] The sum of a geometric progression with infinite terms
The sum S of infinite geometric series will be
Solved Examples for Series Formula
Q: Find the sum of the following series:
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) ].
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 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 +…..
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
| Arithmetic Progression | Geometric Progression | |
| Sequence | a, a+d, a+2d,……,a+(n-1)d,…. | a, ar, ar2,….,ar(n-1),… |
| Common Difference or Ratio | Successive term – Preceding term Common difference = d = a2 – a1 | Successive term/Preceding term Common ratio = r = ar(n-1)/ar(n-2) |
| General Term (nth Term) | an = a + (n-1)d | an = ar(n-1) |
| nth term from the last term | an = l – (n-1)d | an = l/r(n-1) |
| Sum of first n terms | sn = n/2(2a + (n-1)d) | sn = a(1 – rn)/(1 – r) if |r| < 1 sn = a(rn -1)/(r – 1) if |r| > 1 |
*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.
| Sequences | Series |
| Set of elements that follow a pattern | Sum of elements of the sequence |
| Order of elements is important | Order of elements is not so important |
| Finite sequence: 1,2,3,4,5 | Finite series: 1+2+3+4+5 |
| Infinite sequence: 1,2,3,4,…… | Infinite Series: 1+2+3+4+…… |
Sequence and Series Examples
Question 1: If 4,7,10,13,16,19,22……is a sequence, Find:
- Common difference
- nth term
- 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.
Frequently Asked Questions
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+ ..
Arithmetic Series Formula
The word series implies sum. We can transform a given arithmetic sequence into an arithmetic series by adding the terms of the sequence. The example below highlights the difference between the two.
Sequence versus Series
Arithmetic Sequence (list):
2,4,6,8,10,…
Arithmetic Series (sum):
Notice that in a sequence, we list the terms separated by commas while in a series, the terms are added as indicated by the plus symbols.
Therefore, an arithmetic series is simply the sum of the terms of an arithmetic sequence. More specifically, the sum of the first n terms in an arithmetic sequence is called the partial sum. The partial sum is denoted by the symbol Sn.
Below is the general form of the arithmetic series formula. This works best if the first and the last terms are given in the problem.
Notes:
▶︎ The Arithmetic Series Formula is also known as the Partial Sum Formula.
▶︎ The Partial Sum Formula can be described in words as the product of the average of the first and the last terms and the total number of terms in the sum.
▶︎ The Arithmetic Sequence Formula is incorporated/embedded in the Partial Sum Formula. It is in fact the nth term or the last term an in the formula.
▶︎ Become familiar with both the arithmetic series formula and the arithmetic sequence formula (nth term formula) because they go hand in hand when solving many problems.
Before we start working with examples, you may recall me mentioning that the arithmetic sequence formula is embedded in the arithmetic series formula. If we substitute and expand the nth term formula within the partial sum formula, what we will get is a new and useful form of the arithmetic series formula.
Below is the alternative formula for the arithmetic series. Consider this if the last term is not given.
Alternative Arithmetic Series Formula
where:
a1 is the first term
d is the common difference
n is the number of terms in the sum
Examples of Applying the Arithmetic Series Formula
Example 1: Find the sum of the first 100 natural numbers.
This is an easy problem. The purpose of this problem is to serve as an introductory example. This should help you get familiar quickly with the arithmetic series formula. Once you have a basic understanding of how to use the formula, you should be able to tackle more demanding problems as you will see later in this lesson.
Recall that the natural numbers are the counting numbers. We can write the finite arithmetic sequence as
1,2,3,4,…,100
and its related arithmetic series as
1+2+3+4+…+100
Clearly, the first term is 1, the last term is 100, and the number of terms being added is also 1000.
Substitute the values into the formula then simplify to get the sum.
If this is your first time solving this type of problem, you may find this a bit overwhelming. Not that it is difficult but because the values that you need are not explicitly given. This can throw you off because you don’t even know how to get started. However, if you have a strategy from the beginning you will realize this problem is not that bad.
What we need to do is to examine the given series. Identify the values that are pertinent and useful to us. Sometimes by doing it this way, the next logical step will be revealed to us.
I hope at this time, you can agree with me that we have no other option but to use the nth term formula to find n. Once we find the value for n, we will substitute that into the arithmetic series formula together with the first and last terms to find the sum of the given arithmetic series.
So here is the information we have gathered. It means the nth term is what we are looking for.
By knowing the first term, and the common difference of a sequence, we can put together the formula that can determine any term in the sequence.
Series
The series of a sequence is the sum of the sequence to a certain number of terms. It is often written as Sn. So if the sequence is 2, 4, 6, 8, 10, … , the sum to 3 terms = S3 = 2 + 4 + 6 = 12.
The Sigma Notation
The Greek capital sigma, written S, is usually used to represent the sum of a sequence. This is best explained using an example:
This means replace the r in the expression by 1 and write down what you get. Then replace r by 2 and write down what you get. Keep doing this until you get to 4, since this is the number above the S. Now add up all of the term that you have written down.
This sum is therefore equal to 3×1 + 3×2 + 3×3 + 3×4 = 3 + 6 + 9 + 12 = 30.
3
S 3r + 2
r = 1
This is equal to:
(3×1 + 2) + (3×2 + 2) + (3×3 + 2) = 24 .
The General Case
n
S Ur
r = 1
This is the general case. For the sequence Ur, this means the sum of the terms obtained by substituting in 1, 2, 3,… up to and including n in turn for r in Ur. In the above example, Ur = 3r + 2 and n = 3.
Arithmetic Progressions
An arithmetic progression is a sequence where each term is a certain number larger than the previous term. The terms in the sequence are said to increase by a common difference, d.
For example: 3, 5, 7, 9, 11, is an arithmetic progression where d = 2. The nth term of this sequence is 2n + 1 .
In general, the nth term of an arithmetic progression, with first term a and common difference d, is: a + (n – 1)d . So for the sequence 3, 5, 7, 9, … Un = 3 + 2(n – 1) = 2n + 1, which we already knew.
The sum to n terms of an arithmetic progression
This is given by:
- Sn = ½ n [ 2a + (n – 1)d ]
You may need to be able to prove this formula. It is derived as follows:
The sum to n terms is given by:
Sn = a + (a + d) + (a + 2d) + … + (a + (n – 1)d) (1)
If we write this out backwards, we get:
Sn = (a + (n – 1)d) + (a + (n – 2)d) + … + a (2)
Now let’s add (1) and (2):
2Sn = [2a + (n – 1)d] + [2a + (n – 1)d] + … + [2a + (n – 1)d]
So Sn = ½ n [2a + (n – 1)d]
Example
Sum the first 20 terms of the sequence: 1, 3, 5, 7, 9, … (i.e. the first 20 odd numbers).
S20 = ½ (20) [ 2 × 1 + (20 – 1)×2 ]
= 10[ 2 + 19 × 2]
= 10[ 40 ]
= 400
Geometric Progressions
A geometric progression is a sequence where each term is r times larger than the previous term. r is known as the common ratio of the sequence. The nth term of a geometric progression, where a is the first term and r is the common ratio, is:
- arn-1
For example, in the following geometric progression, the first term is 1, and the common ratio is 2:
1, 2, 4, 8, 16, …
The nth term is therefore 2n-1
The sum of a geometric progression
The sum of the first n terms of a geometric progression is:
- a(1 – rn )
1 – r
We can prove this as follows:
Sn = a + ar + ar2 + … + arn-1 (1)
Multiplying by r:
rSn = ar + ar2 + … + arn (2)
(1) – (2) gives us:
Sn(1 – r) = a – arn (since all the other terms cancel)
And so we get the formula above if we divide through by 1 – r .
Example
What is the sum of the first 5 terms of the following geometric progression: 2, 4, 8, 16, 32 ?
S5 = 2( 1 – 25)
1 – 2
= 2( 1 – 32)
-1
= 62
The sum to infinity of a geometric progression
In geometric progressions where |r| < 1 (in other words where r is less than 1 and greater than –1), the sum of the sequence as n tends to infinity approaches a value. In other words, if you keep adding together the terms of the sequence forever, you will get a finite value. This value is equal to:
Example
Find the sum to infinity of the following sequence:
| 1 | , | 1 | , | 1 | , | 1 | , | 1 | , | 1 | , | … |
| 2 | 4 | 8 | 16 | 32 | 64 |
Here, a = 1/2 and r = 1/2
Therefore, the sum to infinity is 0.5/0.5 = 1 .
So every time you add another term to the above sequence, the result gets closer and closer to 1.
Harder Example
The first, second and fifth terms of an arithmetic progression are the first three terms of a geometric progression. The third term of the arithmetic progression is 5. Find the 2 possible values for the fourth term of the geometric progression.
The first term of the arithmetic progression is: a
The second term is: a + d
The fifth term is: a + 4d
So the first three terms of the geometric progression are a, a + d and a + 4d .
In a geometric progression, there is a common ratio. So the ratio of the second term to the first term is equal to the ratio of the third term to the second term. So:
a + d = a + 4d
a a + d
(a + d)(a + d) = a(a + 4d)
a² + 2ad + d² = a² + 4ad
d² – 2ad = 0
d(d – 2a) = 0
therefore d = 0 or d = 2a
The common ratio of the geometric progression, r, is equal to (a + d)/a
Therefore, if d = 0, r = 1
If d = 2a, r = 3a/a = 3
So the common ratio of the geometric progression is either 1 or 3 .
We are told that the third term of the arithmetic progression is 5. So a + 2d = 5 . Therefore, when d = 0, a = 5 and when d = 2a, a = 1 .
So the first term of the arithmetic progression (which is equal to the first term of the geometric progression) is either 5 or 1.
Therefore, when d = 0, a = 5 and r = 1. In this case, the geometric progression is 5, 5, 5, 5, …. and so the fourth term is 5.When d = 2a, r = 3 and a = 1, so the geometric progression is 1, 3, 9, 27, … and so the fourth term is 27.
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