• views: 159
• date: 29 March 2026
• posted by: voska89

What you'll learn

Introduction
Motivation
The Principle of Mathematical Induction
Binomial theorem
Some important observations
Some particular cases
The pth term from the end
Middle terms
Binomial coefficient

Requirements

Basic knowledge of mathematics of 9th and 10th std Mathematics

Description

Principle of Mathematical Induction
- Process of the proof by induction −
- Motivating the application of the method by looking at natural numbers as the least inductive subset of real numbers
- The principle of mathematical induction and simple applications
Binomial Theorem
- History
- Statement and proof of the binomial theorem for positive integral indices
- Pascal's triangle
- General and middle term in binomial expansion
- Simple applications
SUMMARY
Principle of Mathematical Induction
1. One key basis for mathematical thinking is deductive reasoning. In contrast to deduction, inductive reasoning depends on working with different cases and developing a conjecture by observing incidences till we have observed each and every case. Thus, in simple language we can say the word 'induction' means the generalisation from particular cases or facts.
2. The principle of mathematical induction is one such tool which can be used to prove a wide variety of mathematical statements. Each such statement is assumed as P(n) associated with positive integer n, for which the correctness for the case n = 1 is examined. Then assuming the truth of P(k) for some positive integer k, the truth of P (k+1) is established.
3. Property (i) - is simply a statement of fact. There may be situations when a statement is true for all n ≥ 4. In this case, step 1 will start from n = 4 and we shall verify the result for n = 4, i.e., P(4).
4. Property (ii) - is a conditional property. It does not assert that the given statement is true for n = k, but only that if it is true for n = k, then it is also true for n = k +1.
Binomial Theorem
1. A triangle with 1 at the top vertex and running down the two slanting sides. This array of numbers is known as Pascal's triangle, after the name of French mathematician Blaise Pascal. It is also known as Meru Prastara by Pingla.
2. The coefficients nCr occuring in the binomial theorem are known as binomial coefficients.
3. There are (n+1) terms in the expansion of (a+b) n, i.e., one more than the index.
4. In the successive terms of the expansion the index of a goes on decreasing by unity. It is n in the first term, (n–1) in the second term, and so on ending with zero in the last term. At the same time the index of b increases by unity, starting with zero in the first term, 1 in the second and so on ending with n in the last term.
5. In the expansion of (a+b) raise to n , the sum of the indices of a and b is n + 0 = n in the first term, (n – 1) + 1 = n in the second term and so on 0 + n = n in the last term. Thus, it can be seen that the sum of the indices of a and b is n in every term of the expansion.

Who this course is for

Complete Mathematics for Engineering Entrance Exam Preparation. ( IIT-JEE Main | Advanced | BITSAT | SAT | etc.)
Those preparing for board and competitive exams State Board, CBSE, ICSE , IGCSE, MHT-CET & NEET
Courses are suitable for 160 countries from Europe, America, Middle East, Asia, Africa and APAC. Notably England, Germany, France, Sweden, Ireland, Scotland, USA, Canada, UAE, Saudi, Qatar, Kuwait, Malaysia, Indonesia, Myanmar, Newzealand, Australia, South Africa, South Korea, Nigeria, Nepal, Sri Lanka, etc




   
   
   

Forex Trading

Crypto Trading. 20 USD Bonus after KYC

We need your support!
Make a donation to help us stay online
        
Bitcoin (BTC)
bc1q08g9d22cxkawsjlf8etuek2pc9n2a3hs4cdrld
	
Bitcoin Cash (BCH)
qqvwexzhvgauxq2apgc4j0ewvcak6hh6lsnzmvtkem

Ethereum (ETH)
0xb55513D2c91A6e3c497621644ec99e206CDaf239

Litecoin (LTC)
ltc1qt6g2trfv9tjs4qj68sqc4uf0ukvc9jpnsyt59u

USDT (ERC20)
0xb55513D2c91A6e3c497621644ec99e206CDaf239

USDT (TRC20)
TYdPNrz7v1P9riWBWZ317oBgJueheGjATm

0 1 2 3 4 5

| Print version |

	Information
		Users of GUESTS are not allowed to comment this publication.