Principle of Mathematical Induction

mathematical induction, prove inequalities using mathematical induction examples with solution, divisibility statements, summation identities

The Principle of Mathematical Induction:

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

(i) The statement is true for n = 1, that is, 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, that is, the truth of P(k) implies the truth of P(k+1).

Then,  P(n) is true for all natural numbers n.

Working Rule:

Step 1. Verify the result for n = 1.

Step 2. Assume the result to be true for n=k and then prove that it is true for n=k+1.

There are many mathematical results that can be proven using mathematical induction. For example, summation identities, divisibility statements, inequalities, etc.

Examples:


गणितीय आगमन (प्रेरण) का सिद्धांत (Principle of Mathematical Induction in Hindi):

मान लीजिए कि प्राकृत संख्या n से सम्बद्ध एक दिया गया कथन P(n) इस प्रकार है, कि

(i) कथन n = 1 के लिए सत्य है, अर्थात P(1) सत्य है, (अथवा कथन किसी निश्चित प्राकृत संख्या n के लिए सत्य है), और

(ii) यदि कथन n = k (जहाँ k कोई धनात्मक पूर्णांक है) के लिए सत्य है, तो कथन n = k + 1 के लिए भी सत्य है, अर्थात् P(k) का सत्य P(k+1) की सत्यता को दर्शाता है।

तब, सभी प्राकृत संख्या n के लिए कथन P(n) सत्य है।

Working Rule:

Step 1. सबसे पहले n = 1 के लिए परिणाम सत्यापित करें, अर्थात कथन P(1) को सत्यापित करें, (अथवा कथन को किसी निश्चित प्राकृत संख्या के लिए सत्यापित करें)। 

Step 2. मान लें कि परिणाम n=k के लिए सही है और फिर साबित करें कि यह n=k+1 के लिए सही है।

ऐसे कई गणितीय परिणाम हैं जिन्हें गणितीय प्रेरण (mathematical induction) का उपयोग करके सिद्ध किया जा सकता है। उदाहरण के लिए: योग सर्वसमिकाएँ (summation identities), विभाज्यता कथन (divisibility statements), असमानताएँ (inequalities), आदि।

Examples:



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About Lata Agarwal 270 Articles
M.Phil in Mathematics, skilled in MS Office, MathType, Ti-83, Internet, etc., and Teaching with strong education professional. Passionate teacher and loves math. Worked as a Assistant Professor for BBA, BCA, BSC(CS & IT), BE, etc. Also, experienced SME (Mathematics) with a demonstrated history of working in the internet industry. Provide the well explained detailed solutions in step-by-step format for different branches of US mathematics textbooks.

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