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3n for alln > 1. 2n + 1Problem 1.2. Prove by induction that Ci1 12...

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3n for alln > 1. 2n + 1Problem 1.2. Prove by induction that Ci1 12

3n for alln > 1. 2n + 1 Problem 1.2. Prove by induction that Ci1 12



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

In this problem of principle of mathematical induction we have to prove Given a statement. Using principle of mathematical induction for all and belongs to natural number. We have given a statement. Let your fan thursday eight one multiply to multiply three plus to multiply three. Multiply by four Plus up to plus and and plus one endless two. Which is equal to and N plus one N plus two. N plus three. Whole divide by four. First we prove basic statement be off one, one is the smallest natural number, so we put in equal to one in L. H. Is. So we have a value equal to one. Multiply by two, multiply by three. So it will make well to six. Now we find our riches putting an equal to one here. So we have one multiplied by two, multiplied by three, multiplied by four, divide by four. So it is also equal to six here. Religion is equal to averages. So this basic statement is true. Now we find payoff K for the Cape Town. So it will be well too. In a statement one multiplied by two. Multiply by three. Plus to multiply by three. Multiplied by four up to plus. This will be equal to OK. K plus one. And K plus two in our age is it will be okay, K plus one. K plus two and K plus three. And hold divide by four. Now we have to prove okay plus one is equal to R. A. Chase when N equal to K plus one. So we write it be of gay plus okay +12 term. So it will be well do K plus one. K plus two and gay plus three. And we have value of P. R. K. Which is K. K plus one. K plus two. K plus three, divide by four. Supporting here we have this LHs okay, K plus one. K plus two. K plus three. Holy White by four. And we have K plus one. K plus two and K plus three. Now we take common. So common will be K plus one. K plus two and K plus three in the back. It we helped A by four plus one, solving the brigade. We have value equal to K plus one. K plus two. K plus three. And this will be K plus four and hold the right by four. This is our alleges Now we find our ages putting N equal to K plus one, putting and equal to K plus one. We have value of our chase is equal to Okay plus what? In place of fame and and plus one. So it will be K plus two and plus two. So it will be K plus three. N plus three. So it will be K plus four and hold divide by four. Here, we can see alleges equal to Rh is so we can say given a statement is true for all and belongs to natural numbers. So we write it here, given a statement is true for all and belongs to natural number, and this will be our final answer.

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.

So to prove by induction first have to prove the base case of n equals one. It's our relationship on the left foreign. So four times willing that our equation on the right two times and which is one times end plus one. So one plus one which does end approving for equal before so both sides are equal to the end, equals one case now for the inductive step. So given the n is equal Okay, we're trying to prove the K plus one case. So we have four plus e that it was for and which is K in this case, equal to two times k sk plus one so but not approved for the K plus one case, we have to take this relationship and change it toe work. So we have four. I was eight plus. Stop that plus four k And now we're gonna add the new term to it for times. Capel's one. We do that. So now our left hand side is true for the Capel's one case. The right hand side. Let me have the same expression plus or times K plus one. So I left hand side is good. Now we just have to prove that our right hand side works by induction. So there's gonna be equal, take you to case where close to K Earth now plus four K plus four after we distribute. So now, in fact, on a two from everything we Get to Times K Squared was three k plus two, which is factual. You attempts K plus one times came plus two. And now we can rewrite that slightly different to James Capel's one times the quantity of K plus one. Plus. Why did that is because we cannot see very easily that this is just the same with K plus one and substitute for end. So this equation in relationship is true for the inductive case, so are proof is complete.

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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