CBSE Class 11 Maths Revision Notes Chapter 9

Class 11 Mathematics Revision Notes for Chapter-9 Sequences and Series

Class 11 Mathematics Revision Notes for Chapter-9 Sequences and Series has the basic concepts of a sequence, which is an ordered list of numbers and series and the summation of all the elements present. For a sequence, ‘1,2,3,4’ the series would be 1+2+3+4, having the resulting summation of 10. 

These revision notes include a concise explanation of all topics related to Sequences and Series and help students revise these topics from an exam point of view. A quick overview of topics like increasing, decreasing, bounded, convergent, and divergent sequences are provided along with the concepts of A.P and G.P. The formulae for finding the nth term of a sequence is a crucial part of these revision notes. 

Sequence and Series Class 11 Notes

1. Definition:

Sequences are any function with the domain as a set of natural numbers. Real sequences are sequences with the range as a subset of real numbers.

If b1 , b3, b3, b4 …….bn    is a sequence, b1+b2+b3+b4+……+bn  is a series. 

Progressions are when terms of a sequence follow a certain pattern. Terms of a sequence do not necessarily follow patterns. 

The phrase “arithmetic progression” refers to a series of numbers where each term is the sum of the term before it and a set number. If the fixed number is positive, the A.P. is advancing; if it is negative, the A.P. is decreasing.

The fixed term is known as the common difference (d), while the first term of an A.P is denoted by the letter “a.”

Then Nth term of an AP : tn = a + (n-1)d

And d = an – an-1

Sum of the first N terms of an A.P,

Sn = n/2 [ a +  (n-1) d ] = n/2[ a + l ]

where l is the last term of the A.P 

Properties of an AP:

• Increasing, decreasing, multiplying and dividing each term of an AP by non-zero constant results in an AP.
• 3 numbers in an AP: a−d, a, a+d

4 numbers in an AP: a−3d, a−d, a+d, a+3d

5 numbers in an AP: a−2d, a−d, a, a+d, a+2d

6 numbers in an AP: a−5d, a−3d, a−d, a+d, a+3d, a+5d

• An AP can have zero, positive or negative common differences.
• The sum of the two terms of an AP equidistant from the beginning and end is constant and equal to the sum of the first & last terms.
• Any term of an AP (except the first) is equal to half the sum of terms which are equidistant from it.

⇒an=12(an−k+an+k),k<n

tr=Sr−Sr−1

• If three numbers are in AP: a, b, c are in AP ⇒2b=a+c
• Nth term of an AP is a linear expression in n: An+B

where A is the common difference of an AP.

Geometric progression is a sequence where each term is produced by multiplying the previous term by a given (constant) number called the common ratio, which may be calculated by dividing a phrase by its immediately preceding term.

The first term of a G.P. is never 0, because we cannot obtain the common ratio as a real integer by dividing anything by zero.

Let ‘a’ be the first term and ‘r’ be the common ratio,

Then for a G.P : a, ar, ar2, ar3, ar4…….

Nth term of the G.P =  tn= arn−1

Sn = a (1 – rn ) / (1 – r ), r ≠ 1

For an infinite G.P when l r l < 1 and n → ∞

l r l < 1 ⇒ rn   → 0  ⇒ S ∞ = a / 1 – r

Properties of a GP :

• Each term of a GP when multiplied and divided by a non-zero constant yields a GP.
• Terms of GP in the reciprocal form are also GP.
• A GP having 3 consecutive terms: a / r ,a ,ar
• A GP having 4 consecutive terms: a / r2 , a / r, ar, ar2
• If three numbers a, b, and c are in GP then b2  =  ac
• Terms of a GP when raised to the same power also yield a G.P.
• If terms of GP are chosen at regular intervals then this formation yields a GP.
• If the terms equidistant from the beginning and the last are multiplied then their product is always the same and is equal to the product of the first and the last term for a finite GP.
• If a1 , a2 , a3……………an forms GP with non-zero and non-negative terms then log a1 , log a2  , log a3 ……………log an  are in GP or vice versa.

12 + 22 + 32 + ….. + n2

2. Means :

A. Arithmetic mean 

If a, b, c are in AP, the arithmetic mean is b between a and c. 

For n positive terms a1 , a2 , a3 , …… , an in AP,

Arithmetic Mean = a1 + a2 + a3 + ….. + an  / n

• AM between two numbers: 

If a, b are two numbers and a1 , a2 , a3 , …… , an in AP,

Then, 

a1 , a2 , a3 , …… , an  are n AM’s between a and b.

A1 = a + d, A2 = a + 2d, ……. An = a + nd,

d = b – a / n + 1

• Sum of n AMs between a and b is

r =1n Ar= nA 

B. Geometric mean 

If a, b, c are in GP, the b is the geometric mean between a and c. 

b2 = ac or b = ac ; a > 0, b > 0 

• For n GMs between two numbers,

If a, b are two numbers and a, G1 , G2 , G3 , ….., Gn , b are in GP then

G1 = ar, G2 = ar2 , …… Gn  = arn-1,

r = (b/a) 1/ n+1

• Product of n GMs between a and b is 

r = 1nGr = (G)n

C. Arithmetic, Geometric and Harmonic means between two given numbers 

If A, G, H are AM, GM and HM between two integers, a and b

A = a + b /2 

G = ab

H = 2ab / a + b

These have the following properties :

1. A G H
2. G2 = AH which means that A, G, H form a GP
3. ?2 – 2A? + G2 = 0, has a and b as its roots. 
4. If A, G,H are corresponding means between three given numbers a, b and c then equation having a, b, c as its roots is 

?3 – 3A?2 + (3G2 /H)? – G3 = 0 

      5. If A and G are AM and GM between two numbers a and b, then 

a = A + √A2 – G2 , b = A – A√A2 – G2 

3. Sigma Notations:

Theorems are as follows :

• r = 1n(ar + br ) = r = 1n ar + r = 1n br
•   r = 1nka =k  r = 1n ar
•  r = 1nk = nk

4. Sum of n Terms of Some Special Sequences

• Sum of first n natural numbers :

       k = 1n= 1 +2+3+ …….. + n = n(n+1)/2 

• Sum of squares of first n natural numbers :

k=1nk2= 12 + 22 + 33 + ….. + n2 = n(n+1)(2n+1) / 6

• Sum of cubes of first n natural numbers :

k=1n k3 = 13 +23+33+……+ n3 =n(n+1)/22

5. Arithmetico-Geometric series

A progression in which each term can be represented as the product of the terms of an arithmetic progression (AP) and a geometric progression(GP), is called an arithmetico- geometric series. 

If 

AP : 1,3,5, ….. 

GP : 1, ?, ?2, ….

Then, 

AGP : 1, 3?, 5?2 …….

For an AGP,

Sn  = a + (a+d)r + (a+2d)r2 + ….. + [a + (n-1)d] rn-1

Then, Sn = a / (1 – r) + dr (1-rn-1) / (1-r)2 – [a + (n – 1)d] rn / (1 – r) , r ≠ 1

and 

If l r l < 1 and n → ∞ then lim n→∞ = 0

S∞ = a / (1 – r) + dr / (1 – r)2

6. Harmonic Progression (HP)

A harmonic progression is a sequence, reciprocal of whose terms form an AP. 

If a1, a2, a3.….. an  is an HP, 

Then 1 / a1, 1 / a2, 1/ a3 ……. 1/an is an AP or vice versa. 

The sum of n terms of an HP cannot be found using a formula. 

If an HP has ‘a’ as its first term and ‘b’ as its second term, 

Then nth term is tn = ab / b + (n-1)(a-b)

If a, b, and c are in HP, 

b = 2ac / a + c 

or 

a / c = a-b / b-c

7. Harmonic Mean

The harmonic mean between a and c, if a, b, and c are in the harmonic progression is given by b.

b = 2ac / a + c

Sequence and Series Class 11 Notes

Class 11 Mathematics Revision Notes for Chapter-9 Sequences and Series helps students understand harmonic progressions, arithmetic and geometric progressions and the basic concepts of sequences and series. The notes use easy-to-understand language to present the concepts that help students quickly revise this particular chapter. 

Many students find it difficult to remember the formulas of the chapter as the derivations are challenging to understand. However, Class 11 Mathematics Revision Notes for Chapter-9 Sequences and Series by Extramarks help the students to learn proactively by ensuring the accuracy and quality of the notes. 

The topics covered in these notes are as follows:

• Meaning of Sequence
• What Is a Sequence in Mathematics?
• Finite Sequence
• Infinite Sequence
• Types of Sequence
• Arithmetic Sequence
• Geometric Sequence
• Fibonacci Sequence
• Meaning of Series
• Notation of Series
• Finite and Infinite Series
• Types of Series
• Arithmetic Series
• Geometric Series
• Meaning of Geometric Progression (G.P.)
• Meaning of Arithmetic Progression (A.P.)
• Arithmetic Mean
• Geometric Mean
• Relation between A.M. and G.M.
• Special Series
• Sum to n terms of Special Series

Meaning of Sequence

A group of items following a particular pattern with properly listed defined terms as the first term, the second term and so on, is a sequence. 

What Is a Sequence in Mathematics?

A collection of numbers present in an ordered form following a particular pattern is referred to as a sequence. Sequences can be finite (having a limited number of terms) and infinite (having an unlimited number of terms). 

Types of Sequence

The three types of sequences are :

• Arithmetic sequence
• Geometric sequence
• Fibonacci sequence

Arithmetic Sequence

An arithmetic sequence is any sequence in which the difference between each successive term is a constant. It must be in accordance with a fixed number, whether the order is ascending or decreasing.

Geometric Sequence

Any series known as a geometric sequence has a ratio between each next word that can be either rising or falling depending on its constant ratio.

The formulae introduced in Chapter 9 Class 11 Mathematics, have to be learnt and understood. The topics are based on a thorough understanding of the derivations of the formulae. Those looking to maximise their exam preparations should avail of the Revision Notes for Chapter-9 Sequences and Series. The notes break down the equations with a clear understanding of the chapter. 

Tips to Prepare for Exams Using CBSE Sequence and Series Notes

• Students can access the Revision Notes for Chapter-9 Sequences and Series from Extramarks for learning and practising important concepts.
• It is advisable to manage time properly, devote enough time and concentration to each topic, and work on any weaker areas for maximum marks.
• Innovative methods of learning like online quizzes and flashcards, and self-made notes can prove useful for understanding and grasping topics better. 
• Students can practise as many questions to better their technique and to save time during exams. 

Conclusion

Extramarks provides Revision Notes for Class 11 Mathematics Chapter-9 Sequences and Series based on the proper NCERT curriculum based on CBSE guidelines. Subject matter experts have written these notes that help students understand complex concepts and answer to the point. Concepts such as A.P, G.P and harmonic mean are important from the exam point of view and preparation of these topics is crucial for students to score good marks in their examinations.