*Karnataka Class 11 Commerce Maths Principle of Mathematical Induction*

*Karnataka Class 11 Commerce Maths Principle of Mathematical Induction*

*Karnataka Class 11 Commerce Maths Principle of Mathematical Induction : The Karnataka Secondary Education examination board is a state level examination which is conducted in the month of March-April. Also known as the Pre-University Examination, it offers Science, Arts and commerce streams. The question paper is usually of three hours duration and as per the Science stream, it is divided into practical and written. Written examination is purely of subjective category.*

*Karnataka Pre-University board is a government body which organizes the higher secondary examination in the state. The board functions under the Department of Primary & Secondary Education. The board has a total of 1202 government pre-university colleges, 165 unaided Pre-University colleges and about 13 Corporation pre-university colleges.*

*Each year at Karnataka board of pre-university education, about 10 lakh students enroll in the 2 year Pre-University courses. The courses offered by the Pre-University board Karnataka are Humanities (Arts), Science & Commerce. There are 23 subjects, 11 languages and 50 combinations in the Pre-University curriculum. Further, for the academic year 2010-11, the Karnataka Pr-University examination board has 4,43,185 students in Humanities in I & II PUC, 2,47,421 students in Science in I & II PUC & 2,77,189 students in commerce in I & II PUC.*

*Karnataka Class 11 Commerce Maths Principle of Mathematical Induction*

**Tips & Tricks for preparing well and do well in PUC examination:**

**Plan Your Study:**Proper time management and planned study can be beneficial both in exam point of view and for your future growth. Forget all the unnecessary stress and focus on what you can do to succeed. Believe in yourself and plan your study as per your feasibility.**Go Through Previous Years Papers:**Practicing past year papers can help you analyze the portion of the subjects from where questions are frequently asked. Moreover, this can also prove helpful in your preparation.**Mark Important Points:**While studying any topic, make sure you underline the important points so as to recall whenever required.-
**Revision Charts:**This mightsound strange, but it works. Once you have read the topic and have marked the important points, draw out a chart consisting of these important points and stick it on the wall, where you face more often while doing anything. This will help in your revision and in remembering important points.

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*Karnataka Class 11 Commerce Maths Principle of Mathematical Induction*

* To prove certain results or statements in Algebra, that are formulated in terms of n, where n is a natural number, we use a specific technique called principle of mathematical induction (P.M.I) *

*Steps of P.M.I*

* Step I – Let p(n): result or statement formulated in terms of n (given question) *

*Step II – Prove that P(1) is true *

*Step III – Assume that P(k) is true *

*Step IV – Using step III prove that P(k+1) is true *

*Step V – Thus P(1) is true and P(k+1) is true whenever P(k) is true. Hence by P.M.I, P(n) is true for all natural numbers n *

*Karnataka Class 11 Commerce Maths Principle of Mathematical Induction*

*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 problems based on summation only. *

*One key basis for mathematical thinking is deductive reasoning. An informal, and example of deductive reasoning, borrowed from the study of logic, is an argument expressed in three statements: *

*(a) Socrates is a man. *

*(b) All men are mortal, therefore, *

*(c) Socrates is mortal. *

*If statements (a) and (b) are true, then the truth of (c) is established. *

*Karnataka Class 11 Commerce Maths Principle of Mathematical Induction*

*To make this simple mathematical example, we could write: *

*(i) Eight is divisible by two. *

*(ii) Any number divisible by two is an even number, therefore, *

*(iii) Eight is an even number. *

*Thus, deduction in a nutshell is given a statement to be proven, often called a conjecture or a theorem in mathematics, valid deductive steps are derived and a proof may or may not be established, i.e., deduction is the application of a general case to a particular case. *

*Karnataka Class 11 Commerce Maths Principle of Mathematical Induction*

*In contrast to deduction, inductive reasoning depends on working with each case, and developing a conjecture by observing incidences till we have observed each and every case. It is frequently used in mathematics and is a key aspect of scientific reasoning, where collecting and analysing data is the norm. *

*Thus, in simple language, we can say the word induction means the generalisation from particular cases or facts.In algebra or in other discipline of mathematics, there are certain results or statements that are formulated in terms of n, where n is a positive integer. *

*To prove such statements the well-suited principle that is used–based on the specific technique, is known as the principle of mathematical induction. *

*In mathematics, we use a form of complete induction called mathematical induction. To understand the basic principles of mathematical induction, suppose a set of thin rectangular tiles are placed.*

*Karnataka Class 11 Commerce Maths Principle of Mathematical Induction*

*When the first tile is pushed in the indicated direction, all the tiles will fall. To be absolutely sure that all the tiles will fall, it is sufficient to know that *

*(a) The first tile falls, and *

*(b) In the event that any tile falls its successor necessarily falls. *

*This is the underlying principle of mathematical induction. We know, the set of natural numbers N is a special ordered subset of the real numbers. In fact, N is the smallest subset of R with the following property: A set S is said to be an inductive set if 1∈ S and x + 1 ∈ S whenever x ∈ S. Since N is the smallest subset of R which is an inductive set, it follows that any subset of R that is an inductive set must contain N.*

*Karnataka Class 11 Commerce Maths Principle of Mathematical Induction*

*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.*

*Suppose there is a given statement P(n) involving the natural number n such that *

*(i) The statement is true for n = 1, i.e., P(1) is true, and *

*(ii) If the statement is true for n = k (where k is some positive integer), then the statement is also true for n = k + 1, i.e., truth of P(k) implies the truth of P (k + 1). Then, P(n) is true for all natural numbers n. *

*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). *

*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. So, to prove that the property holds , only prove that conditional proposition: If the statement is true for n = k, then it is also true for n = k + 1. This is sometimes referred to as the inductive step. *

*The assumption that the given statement is true for n = k in this inductive step is called the inductive hypothesis. For example, frequently in mathematics, a formula will be discovered that appears to fit a pattern like 1 = 12 =1 4 = 22 = 1 + 3 9 = 32 = 1 + 3 + 5 16 = 42 = 1 + 3 + 5 + 7, etc. *

*It is worth to be noted that the sum of the first two odd natural numbers is the square of second natural number, sum of the first three odd natural numbers is the square of third natural number and so on.Thus, from this pattern it appears that 1 + 3 + 5 + 7 + … + (2n – 1) = n2 , i.e, the sum of the first n odd natural numbers is the square of n. *

*Karnataka Class 11 Commerce Maths Principle of Mathematical Induction*

*Let us write P(n): 1 + 3 + 5 + 7 + … + (2n – 1) = n2 . We wish to prove that P(n) is true for all n. The first step in a proof that uses mathematical induction is to prove that P (1) is true. This step is called the basic step. Obviously 1 = 12 , i.e., P(1) is true. *

*The next step is called the inductive step. Here, we suppose that P (k) is true for somepositive integer k and we need to prove that P (k + 1) is true. Since P (k) is true, we have 1 + 3 + 5 + 7 + … + (2k – 1) = k2 … *

*(1) Consider 1 + 3 + 5 + 7 + … + (2k – 1) + {2(k +1) – 1} … (2) = k2 + (2k + 1) = (k + 1)2 [Using (1)] Therefore, P (k + 1) is true and the inductive proof is now completed. Hence P(n) is true for all natural numbers n.*

*Karnataka Class 11 Commerce Maths Principle of Mathematical Induction*

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