Problem

Prove the following by using the principle of mat…

06:25

Problem 21 Easy Difficulty

Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.
$$
x^{2 n}-y^{2 n} \text { is divisible by } x+y \text { . }
$$

Answer

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Precalculus

NCERT Class 11 - Math

Chapter 4

Principle of Mathematical Induction

Section 1

Introduction

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Video Transcript

in this problem of mathematical induction we have to prove. Given a statement using principle of mathematical induction for all. And belongs to national Number First. Given a statement as we often and we have given a statement X. To the power to a miner's. Why do the power to N. Is divisible by? Yeah X plus. What? First we prove this statement for the of one cooking value and re hell X squared minus Y. Is there. And we know the formula for E. X plus Y and X minus Y. And it is visible by X plus Y. So it is true for the given statement. Now we consider be off. Okay so we help extra the power to k minus Y. To the power to okay we consider this is equal to X plus Y. Multiply by M. Where M is any natural number. Mhm. Now we grew for p. of gay plus one. So it will be extradited. Power to game plus two minus Y. To the power Who gave plus two. We can write this. Power is extra the power to K. Multiplying exist with minus Y. To the power to K. Multiplying wise work. And we realize these sentences extra the power to K excess square minus access. Where Y. To the power to K plus access. Where why do the power to K minus? Why do the power to get multiplying wise work. Taking common from this? two times we held access we're in brigade X. To the power to K minus Y. To the power to K. And taking common from this. Two times we have Y to the power to K. In brigade access square minus Y squared. And here we have value for extra power to k minus Y. To the power to okay X plus Y. Multiplied by game. So we right here I am X plus Y. His excess square plus why to the power to weaken? Writing X plus Y, X minus Y. Now we take common X plus Y from this too. So we have value X plus Y. We didn't brigade came access. Where plus why do the power to game are deploying X minus Y. So we can see here. Considering this whole time I was killed. We have my people of X plus. Why here Q. Is any natural number? Mhm. So we can say given a state maine is to for all. Thank you. Yeah. Mhm. For all. And belongs to nation number. And this is our final answer.

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