Problem

Prove the following by using the principle of mat…

05:09

Problem 8 Easy Difficulty

Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.
$$
1.2+2.2^{2}+3.2^{3}+\ldots+n .2^{n}=(n-1) 2^{n+1}+2 .
$$

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 state wayne. Using principle of mathematical induction for all and belongs to natural number firstly consider given a statement be are fine, which is one. Multiply by two plus to multiply by two square plus. Team are deployed by who is square plus up to and multiply by total the power. And here will be Thank you in here increasing the power of two. And this all equal to in minus one. In minus one. Total the power and plus one plus two. First we proof beyond one which is basically a statement and one is the smallest natural number. First we take alleges and equal to one. So it will be equal to one. Multiply by two. So this is equal to two. Now we take our riches putting an equal to one here. So the zero multiplied by total the power one plus two. So it will be also, it may be due to here alleges equal to our judges. So it is true for the given statement. Now we consider be of K. For the Cape Town. So we write given a statement, I have to get tom one multiplied by two plus two. Multiplied by two square and three multiplied by Tokyo. Up to okay multiply by talking to the power key. So it will be well too K minus one. Go to the power K plus one plus two. Now we have to prove. Mhm. Be off K plus one. So we write it be of gay plus K plus wanted them. So gay plus one. Total the power K plus one. We have value for P R. K which is K minus one. To do the power K plus one plus two. We write it here. This is the largest part. Oh you show Allegis part K minus one. Total the power K plus one plus two. And this is K plus one. Total the power gay plus one. Now we take common from these two terms. So it will be well due to K plus one, investigate k minus one and this will be K plus one. This plus two will be as it is after simplifying it will be equal to. Okay. So we right using exponent law it will be well two K. Go to the power K plus two plus two. Again, we're simplifying and Rite Aid. K plus one minus one. Record close to to the power K plus two plus two. Now we take our just putting and equal to K plus one. So first I write are just part. It will be well too and minus one. Two to the power N plus one plus two. 14. Okay plus one in place. Often we have K plus one minus one. And this will be total the power K plus one plus one. So it will be K plus two plus two. Here we can see l h is equal to marriages. Hence we can say given a statement, it's true for all and belongs to natural number. And this will be over final answer.

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