Sequences and Series: Maths Class 11 Chapter-9
Key Features of NCERT Material for Class 11 Maths Chapter 9 – Sequences and Series
Quick revision notes
In Chapter 8 of Class 11 : you must have learnt about Binomial Theorem . In chapter 9: you will learn about sequences and series.
- Sequence: Sequence is a function whose domain is a subset of natural numbers. It represents the images of 1, 2, 3,… ,n, as f1, f2, f3, …., fn, where fn= f(n).
- Real Sequence: A sequence whose range is a subset of R is called a real sequence.
- Series: If a1, a2, a3, … , anis a sequence, then the expression a1 + a2 + a3 + … + an is a series.
- Progression: A sequence whose terms follow certain rule is called a progression.
- Finite Series: A series having finite number of terms is called finite series.
- Infinite Series: A series having infinite number of terms is called infinite series.
A sequence in which the difference of two consecutive terms is constant, is called Arithmetic Progression (AP).
Properties of Arithmetic Progression
(i) If a sequence is an AP, then its nth term is a linear expression in n, i.e., its nth term is given by An + B, where A and B are constants and A = common difference.
(ii) nth Term of an AP If a is the first term, d is the common difference and / is the last term of an AP, then
(a) nth term is given by 1= an = a + (n – 1)d
(b) nth term of an AP from the last term is a’n = l – (n – 1)d
(c) an + a’n = a + 1
i.e., nth term from the start + nth term from the end
= first term + last term
(d) Common difference of an AP
d = Tn – Tn-1, ∀ n > 1
(e) Tn = 1/2[Tn-k + Tn+k], k < n
(iii) If a constant is added or subtracted from each term of an AP, then the resulting sequence is an AP with same common difference.
(iv) If each term of an AP is multiplied or divided by a non-zero constant k, then the resulting sequence is also an AP, with common difference kd or d/k where d = common difference.
(v) If an, an+1 and an+2 are three consecutive terms of an AP, then 2an+1 = an + an+2.
(vi) (a) Any three terms of an AP can be taken as a – d, a, a + d.
(b) Any four terms of an AP can be taken as a-3d,a- d, a + d, a + 3d.
(c) Any five terms of an AP can be taken as a-2d,a – d, a, a + d, a + 2d.
(vii) Sum of n Terms of an AP
(a) Sum of n terms of AP, is given by Sn = n/2[2a + (n – 1)d] = n/2[a + l]
(b) A sequence is an AP, iff the sum of n terms is of the form An2 + Bn, where A and B are constants. Common difference in such case will be 2A.
(c) Tn = Sn – Sn-1
A progression of numbers arranged in a clear request as per a given certain standard is called sequences. A sequence is either limited or infinite relying on the quantity of terms in a sequences.
On the off chance that a1, a2, a3,… … an is a sequence, at that point the articulation a1 + a2 + a3 + a4 + … + an is called arrangement.
A sequences whose terms follow certain examples are all the more regularly called progression.
Math Progression (AP)
A sequence wherein the distinction of two sequential terms is steady, is called Arithmetic progression (AP).
Properties of Arithmetic Progression (AP)
On the off chance that a sequence is an A.P. at that point its nth term is a linear articulation in n for example its nth term is given by A + B, where An and S are steady and An is basic contrast.
nth term of an AP : If an is the main term, d is normal contrast and l is the last term of an AP at that point
- nth term is stated by an = a + (n – 1)d.
- nth term of the said AP from the last term is a’n =an – (n – 1)d.
- an + a’n = constant
- Common difference of an AP i.e. d = an – an-1,∀ n > 1.
In the event that a consistent is included or deducted from each term of an AR then the subsequent sequence is an AP with same regular distinction.
In the event that each term of an AP is duplicated or separated by a non-zero steady, at that point the subsequent sequence is additionally an AP.
In the event that a, b and c are three successive terms of an A.P then 2b = a + c.
Any three terms of an AP can be taken as (a – d), an, (a + d) and any four terms of an AP can be taken as (a – 3d), (a – d), (a + d), (a + 3d)
Sum of n Terms of an AP
Sum of n terms of an AP is stated by
Sn = [2a + (n – 1)d] = (a1+ an)
A sequence is an AP If the sum of n terms is of the form An2 + Bn, where An and B are steady and A = half of regular distinction for example 2A = d.
a =Sn – Sn-1
If a, A and b are in A.P then A = is known as the arithmetic mean of a and b.
If a1, a2, a3,……an are n numbers, at that point their arithmetic mean is given by
Geometric Progression (GP)
A sequence where the proportion of two successive terms is steady is called geometric progression. The consistent proportion is called normal ratio(r).
for example r = , ∀ n > 1
Properties of Geometric Progression
On the off chance that an is the main term and r is the regular proportion, at that point the overall term or nth term of GP is a =arn-1
nth term of a GP from the last is a’n = , l = last term
In the event that all the terms of GP be multiplied or divided by same non-zero steady, at that point the subsequent sequence is a GP with a similar regular proportion.
The reciprocal terms of the given GP forms a GP.
On the off chance that each term of a GP be raised to some power, the subsequent sequence additionally forms a GP
In the event that a, b and c are three continuous terms of a GP then b2 = air conditioning.
Any three terms can be taken in GP as , a and ar and any four terms can be taken in GP as , , ar and ar3.
Sum of n Terms of a G.P
Geometric Mean (GM)
On the off chance that a, G and b are in GR then G is known as the geometric mean of an and b and is given by G = √(ab).
In the event that a,G1, G2, G3,… .. Gn, b are in GP then G1, G2, G3,… … Gn are in GM’s among an and b, at that point
common ratio r =
In the event that a1, a2, a3,…, an are n numbers are non-zero and non-negative, at that point their GM is given by
GM = (a1 . a2 . a3 …an)1/n
Product of n GM is G1 × G2 × G3 ×… × Gn =Gn =
Important Results on the Sum of Special Sequences
Sum of first n natural numbers is
Σn = 1 + 2 + 3 +… + n =
Sum of squares of first n natural numbers is
Σn2 = 12 + 22 + 32 + … + n2 =
Sum of cubes of first n natural numbers is
Σn3 = 13 + 23 + 33 + .. + n3 =