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Sequence & Series for |Class 11|
Introduction to Sequences and Series: - "In this chapter, we dive into the fascinating world of sequences and series! Learn about the basic definitions, types of sequences, and understand how a series is formed from a sequence. Whether it's arithmetic or geometric, we'll break it down step by step to make the concepts easy to grasp.
Arithmetic Progression (AP) Explained: - “Learn all about Arithmetic Progression (AP) in this comprehensive chapter. We'll cover the general form of AP, its properties, and important formulas, including how to find the nth term and the sum of n terms.
Geometric Progression (GP) Simplified: - "We explore Geometric Progression (GP) – a key concept in sequences and series. Understand how to find the nth term, the sum of terms, and more.
Sum of Finite & Infinite Series: - "How do you find the sum of a finite series? What about an infinite series? This video covers all the necessary formulas and techniques you need to solve problems on finite and infinite series, including detailed examples to help you practice."
Special Series: - Sum of n Terms of Squares and Cubes.
"Get ready to tackle special series! Learn the formulas for the sum of squares and cubes of the first n natural numbers, and see how these special series are used in various problems.
Practice Problems on Sequence and Series: - “Put your knowledge to the test with a variety of practice problems on sequences and series. In this chapter, we'll solve questions from previous years and other challenging problems to make sure you're fully prepared for any exam question!"

In the Sequence and Series chapter, the topics and formulas you will encounter are crucial for solving various mathematical problems. Here’s a breakdown of the topics along with important formulas:
Sequence: - A sequence is an ordered list of numbers following some rule or pattern.
Types of Sequences: - Arithmetic Progression (AP), Geometric Progression (GP), Harmonic Progression (HP)
Arithmetic Progression (AP): - A sequence where the difference between consecutive terms is constant.
nth term of an AP: - a_n = a + (n-1)d, where `a` is the first term, `d` is the common difference, and `n` is the term number.
Sum of the first n terms of an AP: - S_n = n/2[2a + (n-1)d] = [a + l], where `l` is the last term.
Geometric Progression (GP): - A sequence where the ratio between consecutive terms is constant.
nth term of a GP: - a_n = ar^(n-1), where `a` is the first term, `r` is the common ratio, and `n` is the term number.
Sum of the first n terms of a GP (finite GP): - S_n = (ar^(n-1))/(r-1), where r not equal to 1.
Sum of an infinite GP(for |r| less than 1): - S_∞ = a/(1-r), if, |r| less than 1. Harmonic Progression (HP): - A sequence where the reciprocals of the terms form an arithmetic progression (AP).nth term of an HP: - If a sequence is in HP, then its reciprocal will be in AP. The nth term is given by: - H_n = 1/(a+(n-1)d), where `a` and `d` are the first term and common difference of the corresponding AP.
Relation between AP, GP, and HP: -
If three terms a, b & c are in AP, then: 2b = a + c,
If three terms a, b & c are in GP, then: b^2 = ac
If three terms a, b & c are in HP, then their reciprocals are in AP, i.e.:
2/b = 1/a + 1/c
Sum of Special Series: - These special series are often used in problems involving sums of powers of natural numbers.
Sum of first n natural numbers: - S_n = (n(n+1))/2
Sum of squares of first n natural numbers: - S_n = n(n+1)(2n+1)/6
Sum of cubes of first n natural numbers: - S_n = [n(n+1)/2]^2
Miscellaneous Formulas: -
Arithmetic Mean (AM) between two numbers a & b: - AM = (a+b)/2.
Geometric Mean (GM) between two numbers a & b: - GM = √ab
Important Notes: -
In AP, the terms increase or decrease by a fixed number (common difference).
In GP, the terms multiply or divide by a fixed number (common ratio).
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