Sequence & Series-Definition, Types, Formulas, Solved Examples

Master Chapter: Sequence and Series for Class 11 – ANAND CLASSES 📚

🔢 Boost Your Board & JEE Prep with Neeraj Anand’s Expert Guide! 🔢

Sequence and Series – Definition & Explanation

📌 Sequence:
A sequence is an ordered list of numbers that follow a specific rule or pattern. Each number in the sequence is called a term, and the position of a term is denoted by a subscript (e.g., a₁,a₂,a₃,…).

🔹 Example of a Sequence:

• 2,4,6,8,10…………. (Arithmetic Sequence)
• 3,9,27,81,………….. (Geometric Sequence)

📌 Series:
A series is the sum of the terms of a sequence. If the sequence is finite, the series has a limited number of terms; if the sequence is infinite, the series continues indefinitely.

🔹 Example of a Series:

• Arithmetic Series: 2+4+6+8+10+……….
• Geometric Series: 3+9+27+81+…………..

📌 Types of Sequences & Series:
✅ Arithmetic Progression (AP) – Difference between consecutive terms is constant.
✅ Geometric Progression (GP) – Each term is obtained by multiplying the previous term by a constant ratio.
✅ Harmonic Progression (HP) – Reciprocal of an arithmetic sequence.
✅ Special Series – Sum of squares, sum of cubes, etc.

If a₁, a₂, a₃, a_4, ……. is a sequence, then the corresponding series is given by

S_N = a₁+a₂+a₃ + .. + a_N

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

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:

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 T_n and is given by T_n = a + (n-1)d, where “a” is the first term and d is the common difference.
T_n = 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 T_n and is given by T_n = 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(FAQs) Sequence and Series Chapter For Class 11

Q1

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.

Q2

What are Some of the Common Types of Sequences?

A few popular sequences in maths are:

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

Q3

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……a_n. On the other hand, an infinite sequence is never-ending i.e. a₁, a₂, a₃, a₄, a₅, a₆……a_n…..

Series: In a finite series, a finite number of terms are written like a₁ + a₂ + a₃ + a₄ + a₅ + a_{6 +} ……a_n. In case of an infinite series, the number of elements are not finite i.e. a₁ + a₂ + a₃ + a₄ + a₅ + a_{6 +} ……a_n +_…..

Q4

Give an example of sequence and series.

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

Q5

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.

Q6

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, …

Q7

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, …., ar^n-1, ..

Q8

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 + ….+ ar^n-1+ ..