# SEQUENCES AND SERIES CLASS 11 MATHS

## Summary

By a sequence, we mean an arrangement of number in definite order according to some rule. Also, we define a sequence as a function whose domain is the set of natural numbers or some subsets of the type {1, 2, 3, ....k}. A sequence containing a finite number of terms is called a finite sequence. A sequence is called infinite if it is not a finite sequence.

Let a1, a2, a3, ... be the sequence, then the sum expressed as a1 + a2 + a3 + ... is called series. A series is called finite series if it has got finite number of terms.

An arithmetic progression (A.P.) is a sequence in which terms increase or decrease regularly by the same constant. This constant is called common difference of the A.P. Usually, we denote the first term of A.P. by a, the common difference by d and the last term by l. The general term or the nth term of the A.P. is given by

an= a + (n – 1) d.

The sum Sn of the first n terms of an A.P. is given by
$S_n= \frac{n}{2}(2a + (n - 1) d)$

The arithmetic mean A of any two numbers a and b is given by $\frac{a + b}{2}$ i.e., the sequence a, A, b is in A.P.

A sequence is said to be a geometric progression or G.P., if the ratio of any term to its preceding term is same throughout. This constant factor is called the common ratio. Usually, we denote the first term of a G.P. by a and its common ratio by r. The general or the nth term of G.P. is given by an= arn – 1 The sum Sn of the first n terms of G.P. is given by $$S_n= \frac{a(r^n - 1)}{(r - 1)} = \frac{a(1 - r^n)}{(1 - r)}, {\rm if } r \ne 1$$.

The geometric mean (G.M.) of any two positive numbers a and b is given by $\sqrt{ab}$ i.e., the sequence a, G, b is G.P.

## Lecture - 1

• Meaning of Sequences
• Meaning of General Term (nth term)
• Meaning of Series
• NCERT Exercise 9.1 (Q1 to Q14)
• Fibonacci Sequence

## Lecture - 2

• Meaning and Basics of Arithmetic Progression
• Derivation for nth term (General term) of Arithmetic Progression
• Derivation for Sum to n terms of a Arithmetic Progression
• Relationship between Arithmetic Progression and Arithmetic Mean
• NCERT Exercise 9.2 (Q1 to Q4)

## Lecture - 3

• NCERT Exercise 9.2 (Q5 to Q12)

## Lecture - 4

• NCERT Exercise 9.2 (Q13 to Q18)

## Lecture - 5

• Meaning of Geometric Progression (GP) and its common ratio
• General Term of Geometric Progression (nth term of GP)
• Derivation for Sum to n terms of a GP
• Geometric Mean and its relationship with Geometric Progression
• Rules Comparison of Arithmetic Progression and Geometric Progression
• NCERT Exercise 9.3 (Q1 to Q10)

## Lecture - 6

• NCERT Exercise 9.3 (Q11 to Q18)

## Lecture - 7

• NCERT Exercise 9.3 (Q19 to Q25)

## Lecture - 8

• NCERT Exercise 9.3 (Q26 to Q31)

## Lecture - 9

• Introduction to Special Series
• Sum of first n natural numbers
• Inspirational Story of Mathematician Gauss when he was in elementary school told by Prof. Arvind Gupta
• Sum of Squares of first n natural numbers
• Sum of Cubes of first n natural numbers
• Summary of Lecture

## Lecture - 10

• NCERT Exercise 9.4 (Q1 to Q10)
• NCERT Example 19

## Lecture - 11

• NCERT Miscellaneous Exercise (Q1 to Q10)

## Lecture - 12

• NCERT Miscellaneous Exercise (Q11 to Q17)

## Lecture - 13

• NCERT Miscellaneous Exercise (Q18 to Q24)

## Lecture - 14

• NCERT Miscellaneous Exercise (Q25 to Q32)

## Lecture - 15

• NCERT Miscellaneous Example 21 to Example 24

## Lecture - 16

• Why sum to infinity in a Geometric Progression exists only for r less then 1? Derivation of rule for sum of infinite terms of a G.P.
• NCERT Supplementary Exercise (Q1 to Q6)