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Determine by the method of induction that if 1x0 hence1xk forevery k,study the auxiliary equation inthe interval...

Question

Determine by the method of induction that if 1x0 hence1xk forevery k,study the auxiliary equation inthe interval

Determine by the method of induction that if 1x0 hence 1xk for every k, study the auxiliary equation in the interval



Answers

Prove each formula by induction, if possible. $$2+4+6+\cdots+2 n=n(n+1)$$

Gearbox is equal to one plus some. Okay, from one to infinity in K ham's extra power. Okay, which is equal to one plus, um Okay. From one team in the tea. And here we replace a I three times a month. One us or to K minus one. I'm just okay, this is equal to one plus reacts some and from one t o ky a k minus one. Access to a free man. Swan us ex im some hay from 1 to 20. Well, that's the off a menace. One This part is ICO too. Jia vice on this part is equal to you. One over one, minus flax. So what's my excuse? And we have gearbox Zico, too. One class acts or one minus. Relax. You wanted by one minus reacts you can simply by this term this is You can chew one over one minus wax. We can ride this. You go to Psalm a three year old or to K excuse. Okay, so Okay. It's equal to war is about Kay

The given statement is who less for less. 69 upto to em is equal to and spare. Let's and my school. Let's call it equation. Yeah, we have to go there. The human statement does not satisfies the first condition off. Please develop mathematical induction and it satisfies its second condition. So for N is equal to one. Two is equal toe. Once there last one. Let's do that. ISS two is equal to yeah for which is not true. So first condition. See what one is not satisfied. Let us assume there. The given statement is true for some natural number, and it's equal to yeah, that is two plus or less. Six less to care is equal to yes. Good. Let's get less blue. Let's call the equation. The now we have to prove the human statement or end is equal to Yep, that's one that is to bless. Four months, six less Look last two times scape Miss one is equal to careless. One scared less. Give less, one less toe. Let us consider the question sneaky. Now left hand side off C is equal to two plus four plus six plus okay. Last two times give bliss Well, from equation be Visit the two plus 46 off to okay. Yes, Equal toe. Yes. Yeah. Last. Yeah. Let's toe last two times. Give us one. This is your going toe. Bye. Uh, escape escape. Plus two less multiplying by two. Bigger to give. Plus tau now rearranging their terms you get guess Gap last to give less one less give less, one less You the first three terms These care plus one whole spear loss hair plus one blessed to which is equal toe Dr Hand side Uh, whether see, that is simple. Yes, service, fi But the human statement is not true for all natural numbers because the first condition off the filament Mexican induction is not satisfied.

All right when selling for induction, and we have to solve first case and equals one, which is four times one minus one equal times two and plus one. So four times one minus 13 is equal to one time, two times one plus one, which just transaxle three. So r n plus one case works. Now we're gonna let and okay, and we're gonna try and solve for the end is equal to K plus one case. So we have three plus four. Anything. Three plus seven was a doctor before and my one which, you know Annick. Okay, then that's gonna be equal to okay. Times two K plus one. So now what we're gonna do, we're gonna add the next term to both sides, which right in a second plus okay, was one plus before times K plus one minus one, and that is gonna be equal. Okay, times two k plus one plus four times K plus one minus one. All right, so now our left hand side, which is the side above right here. That's already what we're trying to solve. That's it. The K plus one case. No, we have to make our right hand side, the K plus one case. So, in other words, now I'm gonna distribute everything begin. So we have to Okay. Squared los que most four k plus four minus form going. We get to K squared most five k plus three. So, no, I can factor here. Okay. Plus three times K lost one. You can check that real quick to K Times case K squared to Kate times one plus three times K by K and three times one for positive, three on the outside. And then we could be right this as K plus one. Just switching the terms around times two attempts K plus one minus one. Excuse me plus one. And then if we distribute that to in we get two k plus three in this because our K plus one term for our relationship, it's now that we proved for an equals one term. And then we proved that was equals k. It produces the same relationship which we have in red. Yeah, once again is equal to our Capel's one term, which we have down here. You see that our relationship is true

We're going to use mathematical induction to prove that statement as N is true for every positive into the end. For that we follow these steps in part A will verify is one are people right escape, percival, right escape plus one party. We assume that this case true. And use algebra to change escape to escape ALs one and in party will write the conclusion based on steps A three D. The statement sn is the following 1/1 times four plus 1/4 times seven plus 1/4, seven times stand plus up to 1/3 in minus two times three and plus two is equal to and over three and plus one. So the story per se we prove S one in the case of S one, there is only one term. The sum is some reduces to one term, the first term 1/1, 4 times four. That's because for an equal one. These general term here is 1/3 times one minus two, which is three minus two, which is one. And this factor here is three times one plus one is three plus one is four. So it's one over one times four. Which is this term here. So for an equal one, the last and first term is the same. So we got only one term one of one times four on the right side, we have for an equal 11 over three times one plus one. So we can calculate all things here. We had 1/4 which is one times four. And on the right on the right side we have 1/3 times one is three plus one. And so we have 1/4 equal 1/4 which is true. So we can say that s one is true. Now in part B we write statement Sk which is obtained replacing KN by K In the general statement here. So you get one over uh that's a statement. This K. Okay his statement 1/1 times four plus 1/4 times seven. This 1/7 times 10. Mhm. Mhm. Up to one over three K minus two times three K plus one. Okay, that must be equal to Cape over three K plus one. That statement SK and now we write SK plus one which is obtained by replacing and by K plus one in the general statement here. So we get 1/1 times four plus 1/4 times seven plus 1/7 times 10. Plus up to We had this term here before the last term so his three came and it's two times three K plus one plus the last term abuse. 1/3 times K plus one minus two times three times K plus one plus one. In that sequel to. So we have kept this one here and here down. We had three times K. Bliss one plus one. The statement escape plus one and now in party we assume that? S case true and we try to prove that escape this one is also true to assume that is Gay East true. And that means that the equality given here is true at least 1/1. Times four plus 1/4 10 7. This 1/7 times 10 plus up to 1/3 K minus two. Yeah times three K. Bliss one is equal to K over three K. This one and now you do some algebra here and if this statement is true you will continue to be true. If we had the same term both sides of the equation and we are we're going to add this term here. Both sides of physicality. So the following equality is true. 1/1 times 41 over 14 7. This 1/7 times 10 Up to the term before the last one is three K minus two times three K plus one one over that. Plus these terms here. She's won over three cable. This one minus two times three. Keep this one plus one equal. Yeah three is sorry, K plus one. Oh word. Okay three K plus one. Which is this one here? Plus the term. We are adding both sides. This one plus I think it's not going to be in a space here. So we're ready down here equals K over three K. Bliss one plus T stern on over three times K plus one minus two times three times K plus one. This one. Mm And remember this some here is equal to this term here, which is sK which we're assuming to be true. We are adding both eyes of this equality which is through the same term. And because of that, because we are in the same term, both sides of the equality. This new equality here is also true and now we have that we are going to do some simplifications on the right side. The left side remains the same. So we get one over one times four plus 1/4 times seven plus 1/7 times 10. Up to 1/3 K minus two times three K plus one loss three K plus one minus two times three times K plus one for swan on. Over that equal. Now we recognize here this expression here we can simplify the factors and we get K over three K plus one plus. So this first factor here is three K plus three or minus two is one times. And the other factor here is three K plus three plus one is four. It was clear. And with that we get the left side is the same. What? 1/1 times 41 over four times seven. Mhm. Yeah. Mhm Plus 1/7 plus 10. Plus up to 1/3 k minus two times three K plus juan plus this term here. Three K plus one minus two times three times K plus one. Last one. One over that equal. And in this some here we got the common denominator three K plus one time three K plus four. Yeah. Okay. And if we divide that bye three K plus one, we get three K plus four. That's going to play by K. Here. So we get K times three K plus four plus and three cables. One time three K plus four divided by three cables. One times four is one times this one. Here is one. And now to gain a little bit of time, I'm going to simplified numerator right here and we get three K plus one time through K plus for indeed, uh, denominator and numerator, we get three K square developing these factor here. Inside parenthesis three case square plus four K plus one. Now we okay, can factor out this polynomial in the numerator. It's easy to see that is equal to he seems to see that these three K plus one times K plus one. In fact, if we do these products, we get three K times K, three K square plus three K times +13 K plus K. One times case K plus one times one is one. So we get three K square plus these two terms simplifies simplified to four K plus one. These phenomena here. So it's correct. That's a factory ization of the numerator. And now we can simplify these two, three K plus one in the numerator and denominator. The factor of three K plus one cancel out. And this is equal to gave plus one over three K plus four. And maybe we must write this each of these steps in the in separate implications where I'm going, I'm simplifying a little bit of time here. So 1/1 times four plus 1/4 times seven plus one over seven times 10 plus up to went over three K minus two times three K plus one plus three times K plus one minus two times three times K plus one plus one. One over that equal. Equal, Okay, K plus one over this term here. Three K plus four is the same as three times K plus one plus one. Because he is who he is develop this product here we get three capable three and that plus one is for cable four. So we have uh this expression here which is in fact it's K plus one. If we look at here are C this statement we have here, it's just it's exactly this one and we can then say that S K plus one is also true. If escape is true, escape. This one is also true. So is K plus one is true whenever is K. Oh it's true in that for any positive integer key. Okay, so now in party we're going to combine A plus D. And that implies that S. N. Is true for every was still into the end. Yeah. Mhm. And that is because for a says that this one is true. I'm applying this result here because as one is true, we put K equals one, and we get it as too must be true also. But now again, we applied this same statement with cake was too and then being as to true, you must have the three is true and we can go on forever and we get the result that S. N. The statement is true for every positive into your end. So we have proof by induction that that statement, the given statement is ends true for every positive into your end.


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