By going through these CBSE Class 11 Maths Notes Chapter 9 Sequences and Series Class 11 Notes, students can recall all the concepts quickly.
Sequences and Series Notes Class 11 Maths Chapter 9
Sequence : A set of numbers arranged according to some definite rule is called a sequence.
The different numbers which form the sequence are called the terms of the sequence. Terms are denoted by the symbols T1 T2, T3, … etc. the subscript denotes the position of the term. The nth term is also called the general term of the sequence.
Sequence containing finite number of terms is called a finite sequence and a sequence is called infinite, if it is not a finite sequence.
In a sequence, we should not always expect that the terms of a sequence will be necessarily given by an algebraic formula. For example, in a sequence of prime numbers 2, 3, 5, 7, …, there is no known formula for the prime numbers. But in the sequence, we do have a rule or law for writing its numbers in order.
Series : If the numbers forming the sequence are connected by the signs of addition (+), we get a series.
The numbers of the series are known as the terms of the series.
Progression : If the terms of a sequence constantly increase or decrease in a set pattern, it is called a progression.
Arithmetic Progression (A.P.) : A sequence in which the difference of any term from the previous term is constant, is called an arithmetic progression and the constant difference is called the common difference.
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Notation :
(i) The first term is denoted by a.
(ii) The common difference is denoted by d.
(iii) d = T2 – T1 = T3 – T2 = T4 — T3 = … = Tn — Tn-1
nth term or general term of an A.P.: If a be the first term and d, the common difference of an A.P., then Tn = a + (n – 1 )d.
i.e., nth term = first term + (n – 1) x common difference.
Properties of an A.P.: 1. If a constant is added to each term of an A.P., the resulting sequence is also an A.P.
2. If a constant is subtracted from each term of an A.P., then resulting sequence is also an A.P.
3. If each term of an A.P. is multiplied by a constant, then the resulting sequence is also an A.P.
4. If each term of an A.P. is divided by a non-zero constant, then the resulting sequence is also an A.P.
Sum to n terms of an A.P.
(i) Sn = \(\frac{n}{2}\) (a + 1), where T1 = a, Tn = 1.
(ii) Sn = \(\frac{n}{2}\) [2a + (n – 1)d], where T1, = a, c.d. = d.
Limit of sequence calculator is the value of the series is the limit of the particular sequence.
Notes:
1. Formula (i) is used, when the last term is known and formula (ii) is used when the common difference is known.
2. These formulae have four quantities each. If three are known, the fourth can be found out.
Arithmetic Means : When three quantities are in A.P., the middle quantity is said to be Arithmetic Mean (A.M.) between the other two.
Thus, if, a, A, 6 are in A.P., then A is the A.M. between a and b.
∴ A-a = b-A [∵Each-c..d.]
⇒ 2A – a + b
A = \(\frac{a+b}{2}\)
A.M. between two numbers = Half their sum
An important result: Sum of n A.M.’s between two quantities is n times the single A.M. between them.
n Arithmetic means : Let A1, A2, A3, …, An be the n A.M.’s between two numbers a and b.
The total number of A.M.’s and numbers a and b = (n + 2)
The last term = b.
b = a + (n + 2- 1 )d i.e., d = \(\frac{b-a}{n+1}\)
∴ We can find the A.M.’s as
A1 = a + d = a + \(\frac{b-a}{n+1}\) , A2 = a + 2d = a + \(2\frac{b-a}{n+1}\) ……
An = a + nd = a + \(n\frac{b-a}{n+1}\).
Geometric progression (G.P.): A sequence in which the ratio of any term to the proceeding term is the same throughout, is called a Geometric Progression and the constant ratio is called the common ratio of G.P.
Standard form of G.P. : If a is the first term and r is the common ratio, then a, ar, ar2, … is the standard form of G.P.
nth term or general term of G.P.:
Tn = arn-1>, where T1 = a and c.r. = r.
The above formula has four quantities a, r, n and Tn. Out of these four, if any three are given, fourth can be found out.
Sum of n terms of a G.P.: If G.P. is a, ar, ar2,…; arn-1, then
(i) For, r = 1,
Sn = a + a + a + … + a (n terms)
= na
(ii) For r ≠ 1,
Sn = \(\frac{a\left(1-r^{n}\right)}{1-r}\), if |r| < 1
and Sn =\(\frac{a\left(r^{n}-1\right)}{1-r}\), if |r| > 1
Geometric Means : When three numbers are in G.P., then, middle one is called the Geometric Mean (G.M.) between the other two.
Thus, if a, G, b are in G.P., then G is the G.M. between a and b. It can also be defined as under :
If any number of numbers are in G.P., then all the terms between the first and last terms are called Geometric Means (G.M.’s) between them.
If a, G, b are in G.P. then
\(\frac{\mathrm{G}}{a}=\frac{b}{\mathrm{G}}\)
[v each = common ratio]
G2 = ab ⇒ G = \(\sqrt{a b}\) .
An important result: The product of n G.M.’s between a and b is equal to the nth power of the G.M. between a and b.
n-Geometric Means : Let G1 G2, G3, …, Gn be the n G.M.’s between a and b. Total number of terms = n G.M’s and the numbers a and b = n + 2.
∴ (n + 2)th term = last term = b.
If r is common ratio, b = arn+2-1 + 2 – 1 = arn+1
r = \(\left(\frac{b}{a}\right)^{n \frac{1}{n+1}}\)
G1 = ar = a\(\left(\frac{b}{a}\right)^{\frac{1}{n+1}}\) ,
G2 = ar2 = a\(\left(\frac{b}{a}\right)^{\frac{2}{n+1}}\),……. Gn = arn = a\(\left(\frac{b}{a}\right)^{\frac{n}{n+1}}\)
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Sum of first n natural numbers :
Σn = 1 + 2 + 3 + … + n = \(\frac{n(n+1)}{2}\)
Sum of the squares of the first n natural numbers
Σn2 = 12 + 22 + 32 + … + n2 = \(\frac{n(n+1)(2 n+1)}{6}\)
Sum of the cubes of the first n natural numbers
Σn3 = 13 + 23 + 33 + … + n3 = \(\left[\frac{n(n+1)}{2}\right]^{2}\)
Clearly, Σn3 = (Σn)2
Sum of it terms of a series whose nth term is an3 + bn2 + cn + d :
If Tn = an3 + bn2 + cn + d, then Sn = aΣn3 + bΣn2 4 cΣz + dn.
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