Problem

Prove the following by using the principle of mat…

05:52

Problem 11 Easy Difficulty

Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.
$$
\frac{1}{1.2 .3}+\frac{1}{2.3 .4}+\frac{1}{3.4 .5}+\ldots+\frac{1}{n(n+1)(n+2)}=\frac{n(n+3)}{4(n+1)(n+2)}
$$

Answer

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Precalculus

NCERT Class 11 - Math

Chapter 4

Principle of Mathematical Induction

Section 1

Introduction

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Problem 1

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Problem 10

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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 we laid given a statement be our friend. Yeah. And here we have LHs have given a statement one upon one. Multiply by two. Multiply by three plus one upon to multiply by three. Multiply by four plus one upon three. Multiplied by four. Multiplied by five plus up to plus one upon and N plus one. N plus two. And which is equal to end. Multiplying N plus three are born for multiplying in plus one and N plus two. First we prove for P. F one which is basic statement and one is is molest natural number. So first we take a latest part in equal to one putting here. So we have one upon one. Multiply by two. Multiply by three and this will be equal to 1.6. Now we find averages putting an equal to one here, so we have one multiply by four upon four multiply one plus 12 Mhm. Multiplied by three. This is also equal to 1.6. Here we can see alleges equal to rhs so this gets, the statement is true for the given statement. Now they find P R K. Mhm. Up to get the term for the given statement. So we write one upon one. Multiply by two. Multiply by three plus to multiply by three. Multiplied by four plus one upon three. Multiplied by four, multiplied by five plus up to plus one upon Okay, gay plus one multiplying K plus two. Which is acquired. Okay, K plus three whole upon four K plus one. K plus two. Now we need to prove be of K plus one. Yeah. So we take LHs up to K plus one to term life. Be off K. Abdicate some plus. Okay. Plus 100 will be one upon gay plus one. Multiplying K plus two. Multiplying K plus three. So first we need to find these alleges. Mhm. Mhm. Mhm. So we have the R. K. Y. Two K. Multiplying K plus three. All upon four. K plus one. Multiplying K plus two. And this time one upon K plus one. Multiplying K plus two. Multiplying K plus three. So we take common K plus one and K plus two from the denominator. So we have one upon K plus one. Multiplying K plus two within the decade. We have value. Okay K plus three upon four plus one upon K plus three. So taking L. C. M. And simplify the expression. They help one upon K plus one. Multiplying K plus two within brocade, take calcium. So it will be four K. Plus three. So we cross multiply with K plus three here and here. We multiply by four. First. We expand this K. Plus three. Holy square. Mhm. So we write it one upon K. Plus one. Gay plus two. Okay Plus three and four. Also in denominator nominator. We have value. Okay. And expanding the escape plus three. Holy square K. Square plus nine plus six K. Reporting. Mhm. Platform. Mhm. Again we multiply this. Break it with K. Yeah so we have a value equal to we want to do. Mhm. Thank you. Plus nine K. Plus six K. Square plus four. In denominator we have four multiply K. Plus one. Multiply K plus two. Multiplying K. Plus three. Now the met K. Plus one is square mm arranging the nominator. So we write it que. Que. And this time who case Where? Mhm. Okay. Is good plus. Okay. Okay so rest up tom will be for gay square. And this. Mhm. Four cases where plus eight K. And these four will be as it is in the denominator. Yes. We have four multiplying K plus one K plus two and K plus three. We take common gave from the street. Um So we have K. K. Square plus two K plus one. Here we take common for so we have in brigade K square plus two K plus one. Mhm. We didn't wreck aid. We made eight K plus one. Holy square and denominator all terms will be as it is. Yeah. Okay now we take common. K square plus two K plus one. Holy hell. K square plus two K plus one. Multiplying within brigade K plus four. Yeah. Whole upon four K plus one. K plus two. Multiplying K plus three. And we make it K plus one. Holy square. In this case a plus four will be as it is for K plus one. K plus two. Okay plus three. Now we can divide this K plus one by this. K plus one. So we have value. Mhm. K plus one. Also we arranged the escape plus for like K plus one plus three. In denominator we have value. Okay four. We write it. Mhm. This K plus two ways. K plus one plus one. And this K plus three years. A plus one plus two. This is our LHs part. Now we find or ridges putting an equal to K plus one. In the given statement I saw you even the statement N. N plus three. Four. N plus one and N. Plus who? Mhm. So I write it here and and plus three whole upon for and plus one multiplying and plus two here they put in equal to K plus one. So we have. Okay plus one gay plus one plus three. So it will be like a plus one plus three in the denominator we have four gay plus one last one and one more brigade. Okay. Plus one mm plus two. So we can see here religious equal to our riches. Yeah. So we can right here alleges it will do averages and we can say given a statement and if possible is true for all and belongs to national number. And this will be our final answer.

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