Sequences And Series Class 11th Notes - Free NCERT Class 11 Maths Chapter 9 notes - Download PDF

In our daily life, we often notice patterns, like the number of steps we walk each day increasing by a fixed number of steps on subsequent days, or if the salary of a person is multiplied by a fixed value every year. These are examples of sequences and series. A sequence is simply a list of numbers in a specific order, and a series is the total you get when you add up the numbers in a sequence. NCERT notes for Class 11 Maths Chapter 9 give a detailed explanation of Arithmetic Progression (AP), where the same number is added each time, and Geometric Progression (GP), where each number is multiplied by the same value. You’ll also learn how to find any term in the pattern and how to calculate the total of all terms. The NCERT Class 11 Maths chapter 9 notes are entirely based on the useful topics of the series and sequence.

A Class 11 Maths chapter 9 note helps a student to revise the key concepts before the exam. NCERT Notes for Class 11 Maths chapter 9 covers Arithmetic progression, geometric progression, arithmetic mean, geometric mean, and the relation between the arithmetic mean and geometric mean as per the latest syllabus.

NCERT Class 11 Maths Chapter 9 Notes

Sequence

The general definition is arranging something in a particular order.

Generally, we write the terms of a sequence by $a_1, a_2, a_3, \ldots \ldots \ldots a_n, \ldots \ldots$, etc., the subscripts define the position of the term. The nth term defines the nth position of the sequence.
From the above sentences, we can say that a function whose domain is the set of natural numbers or a subset of it can be defined as a sequence. A functional notation can be used for $a_n$ is a(n)

Series

Let $a_1, a_2, a_3, \ldots \ldots \ldots a_n, \ldots \ldots$, be a given sequence. Then, the expression $a_1+a_2+a_3+\ldots \ldots .+a_n$ is known as a series.

The sum of the following series is denoted by ∑a_n is a(n).

Arithmetic Progression (A.P)

Let the sequence be

$a_1, a_2, a_3, \ldots \ldots a_n$ called an AP when the difference between $a_n$ and $a_n+1$ is $d$
Thus $a_n$ can be written as $a_n=a+(n-1) d$
Let the sequence $a_1+a_2+\ldots. .+a_n$
Then the summation is $S_n=\left(\frac{n}{2}\right)[2 a+(n-1) d]$

Example:

If the 1,5,9,13… sequence is an A.P., what will be the nth term in the sequence?

Solution:

Given, sequence 1,5,9,13 . . . . . . .

Difference between the numbers, d =$(13-9)=4$

The nth term will be,

$\begin{aligned} & a_n=1+(n-1) 4 \\ & \Rightarrow a_n=1+4 n-4 \\ & \Rightarrow a_n=4 n-3\end{aligned}$

Arithmetic Mean

If the two given numbers are p and q. Insert the number F between p and q so that p, F, and q are in arithmetic progression. If F is the arithmetic mean of p and q numbers. Then we will have,

$\begin{aligned} & F-p=q-F \\ & \Rightarrow F+F=p+q \\ & \Rightarrow 2 F=p+q \\ & \Rightarrow F=\frac{p+q}{2}\end{aligned}$

If the terms of an A.P. are increased, decreased, multiplied, or divided by the same constant, they remain in A.P.
If $a_1, a_2, a_3 \ldots$ are in A.P. with common difference $d$, then
(i) $a_1 \pm k, a_2 \pm k, a_3 \pm k, \ldots$ are also in A.P with common difference $d$.
(ii) $a_1 k, a_2 k, a_3 k, \ldots$ are also in A.P with common difference $d k(k \neq 0)$. and $\frac{a_1}{k}, \frac{a_2}{k}, \frac{a_3}{k} \ldots$ are also in A.P. with common difference $\frac{d}{k}(k \neq 0)$.
If $a_1, a_2, a_3 \ldots$ and $b_1, b_2, b_3 \ldots$ are two A.P., then
(i) $a_1 \pm b_1, a_2 \pm b_2, a_3 \pm b_3, \ldots$ are also in A.P
(ii) $a_1 b_1, a_2 b_2, a_3 b_3, \ldots$ and $\frac{a_1}{b_1}, \frac{a_2}{b_2}, \frac{a_3}{b_3}, \ldots$ are not in A.P.

If $a_1, a_2, a_3 \ldots$ and $a_n$ are in A.Ps, then
(i) $a_1+a_n=a_2+a_{n-1}=a_3+a_{n-2}=\ldots$
(ii) $a_r=\frac{a_{r-k}+a_{r+k}}{2} \forall k, 0 \leq k \leq n-r$
(iii) If $n^{\text {th }}$ term of any sequence is linear expression in $n$, then the sequence is an A.P.
(iv) If sum of $n$ terms of any sequence is a quadratic expression in $n$, then sequence is an A.P.

Geometric Progression (G.P)

Let us consider the following sequences:

2, 4, 8, 16, .…..

1/9, -1/27, 1/81, -1/243, ......

Let the sequence be $a_1, a_2, a_3, \ldots \ldots . a_n$ called a GP when the difference in the ratio between $a_n$ and $a_{n+1}$ is $r$ by letting $a_1=a$, we obtain a geometric progression,$a$, ar, ar ${ }^2$, ar ${ }^3, \ldots .$.

General Term Of A G.P

The general term of GP is $a_n=a r^{n-1}$

Sum to $n$ terms of a G.P

$\begin{aligned} & S_n=a+a r+a r^2+\ldots \ldots+a r^{n-1} \\ & \Rightarrow S_n=a \frac{\left(1-r^n\right)}{(1-r)}\end{aligned}$

Geometric Mean (G.M.)

The formula for GM is

G=√ab

(i) If the terms of a G.P. are multiplied or divided by the same non-zero constant $(k \neq 0)$, they remain in G.P.

If $a_1, a_2, a_3, \ldots$, are in G.P., then $a_1 k, a_2 k, a_3 k, \ldots$ and $\frac{a_1}{k}, \frac{a_2}{k}, \frac{a_3}{k}, \ldots$ are also in G.P. with same common ratio, in particularly if $a_1, a_2, a_3, \ldots$ are in G.P., then $\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \ldots$ are also in G.P.
(ii) If $a_1, a_2, a_3, \ldots$ and $b_1, b_2, b_3, \ldots$ are two G.P.s, then $a_1 b_1, a_2 b_2, a_3 b_3, \ldots$ and $\frac{a_1}{b_1}, \frac{a_2}{b_2}, \frac{a_3}{b_3}, \ldots$ are also in G.P.
(iii) If $a_1, a_2, a_3, \ldots$ are in A.P. $\left(a_i>0 \forall i\right)$, then $x^{a_1}, x^{a_2}, x^{a_3}, \ldots$, are in G.P. $(\forall x>0)$

(iv) If $a_1, a_2, a_3, \ldots, a_n$ are in G.P., then $a_1 a_n=a_2 a_{n-1}=a_3 a_{n-2}=\ldots$

Relationship Between A.M. and G.M

A=(a+b)/2 and G=√(ab)

The relation is A ≥ G.

Sum To n Terms Of Special Series

$\begin{aligned} & 1+2+3+\ldots .+n \quad \text { (sum of first } \mathrm{n} \text { natural numbers) } \\ & k=\frac{n(n+1)}{2} \\ & 1^2+2^2+3^2+\ldots .+n^2 \quad \text { (sum of squares of the first } \mathrm{n} \text { natural numbers) } \\ & k=\frac{n(n+1)(2 n+1)}{6} \\ & 1^3+2^3+3^3+\ldots \ldots+n^3 \quad \text { (sum of cubes of the first } \mathrm{n} \text { natural numbers) } \\ & k=\left[\frac{n(n+1)}{2}\right]^2\end{aligned}$

NCERT Class 11 Notes Chapter Wise