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Problem 2. (24 points) Use induction to prove that for any n 2 1, if a > b 2 1 and Euclid(a,b) takes n iterations then a > fn+2 and b > fn+l where fn is th...

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Problem 2. (24 points) Use induction to prove that for any n 2 1, if a > b 2 1 and Euclid(a,b) takes n iterations then a > fn+2 and b > fn+l where fn is the nth term of Fibonacci sequence:hint: Consult with the slides and posted video for the lecture 20.Problem (15 points) Prove the following corollaries where fn is the nth term of Fibonacci sequence. (A Corollary is proposition that follows from proved theorem:) a) Corollary 1: For all n > 1 if a > b 2 1 and b < fn+l then Eucl

Problem 2. (24 points) Use induction to prove that for any n 2 1, if a > b 2 1 and Euclid(a,b) takes n iterations then a > fn+2 and b > fn+l where fn is the nth term of Fibonacci sequence: hint: Consult with the slides and posted video for the lecture 20. Problem (15 points) Prove the following corollaries where fn is the nth term of Fibonacci sequence. (A Corollary is proposition that follows from proved theorem:) a) Corollary 1: For all n > 1 if a > b 2 1 and b < fn+l then Euclid(a,6) takes fewer than iterations_ hint: Consult with the slides and posted video for the lecture 20. Use the theorem in problem to prove Corollary To prove Corollary 1, do not use induction. b) Corollary 2: For all n > 2 if a < fn or b < fn then Euclid(a,b) takes strictly fewer than n iterations hint: Consult with the slides and posted video for the lecture 20. Use the theorem in problem 2 and Corollary 1to prove Corollary 2. To prove Corollary 2, do not use induction.



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Fibonacci Sequence Problem: These numbers form the Fibonacci seque nce: $1,1,2,3,5,8,13,21,34,55, \dots$ PICTURE CANT COPY a. Figure out the recursion pattern followed by these Fibonacci numbers. Write the next two terms of the sequence. Enter the recursion formula into your grapher. You will need to enter $u(n \mathrm{Min})=\{1,1\}$ to show that the first two terms are given. Make a table of Fibonacci numbers and scroll down to find the 20 th term of the sequence. b. Find the first ten ratios, $r_{m}$, of the Fibonacci numbers, where $$r_{n}=\frac{t_{n+1}}{t_{n}}$$ Show that these ratios get closer and closer to the golden ratio, $$r=\frac{\sqrt{5}+1}{2}=1.61803398 \ldots$$ c. Find a pinecone, a pineapple, or a sunflower, or a picture of one of these. Each has sections formed by intersections of two spirals, one in one direction and another in the opposite direction. Count the number of spirals in each direction. What do you notice about these numbers? PICTURE CANT COPY d. Look up Leonardo Fibonacci (also known as Leonardo of Pisa) on the Internet or via another reference source. Find out when and where he lived. See if you can find out how he related the sequence to the growth of a population of rabbits, and why, therefore, his name is attached to the sequence.

To find the relation between end on FN that is written an Antam off few Beneke sequence and we feel forced right after Monica sequence a few times off, Dominica sequence on one one. So three, five it and so on. Here, this is our first time. This is second. This is tour 4 50 six and so on. So here we can see that our relation should be and is less than equals two effin for all and greater than equals two Fine since for unless than five this and is greater than the Energoatom off even a sequence. So this is a inequality. Now we need to prove that this inequality is true for all and greater than equals to five. Now, firstly considered this statement as being that any is left and equals two effort for all and greater than equals to fight No, Let us check in four step. If this statement is true for n equals five since five years, the least value off and so B or five becomes fine is less than 50 m a few minutes the sequence. So if you know that 50 I'm off Monica sequence is five itself This is it them? So therefore are being statement is true for n equals one Now in second step, we need to prove up in second step We assume that our statement is true for unequal skay. So as you dad, being is true Be any other statement which is true for and equals K And this is our induction hypothesis That BK statement So induction I put this is is because statement be true We're statement big it is okay is less than equals two SG gays less than equals toe f k This is a pick A statement So now we want to use this statement to prove that uh, b k plus one is also true So we need to show that statement B k plus one is also true. Therefore, a statement PK plus one is K plus one is less than equals toe fo escape less one no To prove this statement considered left inside off any quality So left prints out of this inequality is K plus one on we can write gay gays less than f k by using induction. I potus is bless one And since we know that if the end one in the SG. It will all always the less than or equals to the next time F K plus one. Since this is a pattern often Monica, sequence. Therefore, this is our event right inside off a question. Therefore we get that K plus one is less than equals toe foo escape less one Therefore, our statement B K plus one four laws from speaker and this completes an induction step. Now, having proved steps one in tow, we conclude that by principal off mathematical induction B m i The statement PN is true for all natural numbers Put all natural numbers greater than equals to five for all natural numbers and they're better than equals to fight. So we have proved our statement being

Begin a statement that says that for N is good than or equal to two, the falling expression is correct. That's one is equal to F and plus one f N f n if and minus one now the actually prove it for every end that belongs to M. That is not full numbers on a square than or equal to tip. So for the logistical base case and is equal to one forced just a generous people not as good. One I'm about my right hand is equal to two and put it, and the expressions that we have see run long 01 two is equal to F three of two of two, enough one. So we have to prove that one one 101110 is equal to 211 one so we can see that when we saw these, we get something like one plus one one plus 01 closed zero on one plus zero, which is equal to to 111 and certified. That's where basis for induction hypotheses. Let's take a musical to take bridges were than to. Now we have to assume that 11 run zero this matrix braced for cake is equal to F K plus one F k Uh, okay. And F K minus one. The actors IAM does now. We're gonna use this to your basis, Allegis. Take. I'm not our base case are induction days that's taken is equal to K plus one. This is what we have to prove to prove the expression. So the evil, whereas something like one mom 10 based power Kate plus one is equal to f K plus two f k plus one f k plus one f k. Now we can break it up into parts and by the house. Month one line zero. Okay, anti 1110 britches. My using election hypotheses. We can take the value of that and put it here, and we get something like F K flus one f k f k ok, minus one multiplied by one. Fun 10 This will do. Is something along lines off f. Okay. Plus one plus f. Okay. On f k plus one plus zero f k plus one plus. Have, uh no, my zero. I'm here to get something like again que plus zero. I first want to miss f came. Plus two. Since these are added, that's gave Lissan escape us on that day. Uh, this right here I'll express is equal to this My here. All right. Just now we have proven that. So this expression

Given the statement, Effin Square plus F to square plus F and square is equal to half and into the F and plus on. We have to prove it using election where and belongs to him that is any night forever. So let's just take our base case and the sequence of on if on denotes different marches sequences until so we get F. One square is equal to F one F two, which is fund basically one into one, which is correct. Now the base is correct so we can move on to the induction hypotheses that then is equal decay. So let's just put Angel's K and White at our left Blohm. Plus after you f K is equal. Teoh f k f K flows along. This is our induction hypotheses that we use later. Let's just look at induction case that is an musical decay, less money so we can be right beside that fun plus F to host Word forgot that F case. Where was F K plus Month square busy full till F. K plus one and two have cake close to. So now we can divide the less insight into two parts that is the force part using induction Que in the knife Odyssey's on the second part. So we get something like F k F K Flex Mud plus F K plus one Hold Square is equal to have k plus one f K plus two so we can take f Think f k plus uncommon here on take it out and get something like F K plus one into F k first f. Hey, 1st 1 Now we know that using the Fibonacci sequence this all equal still f k plus two. So we get something like F K plus one f k plus two. Is he quoted f Okay, plus one f k those two.

We're doing a statement that says that, but we have to make inequality that somebody somewhere along lines of em. It's more than F, and that is the end term off the financial sequence plus two. So we're giving dissonant quality, and we have to prove it. So legis food Using mathematical induction, just they get the face case as and is equal to one. So that is true, because one is slow. That one plus two it's just one is smaller than one plus two, which is one is smarter than three but just proved. Now let's take it on is a good cake. This is our inductive, I thought sees no For this, we have to replace or VSE as you left K. It's more than F se plus to the incision This. So now let's after we have the enders, we have proved or induction case. There's N is equal to K plus one. We have to prove this to prove or an apology. So I just like that first escape bliss on this smaller than F K plus one plus two. We have to prove this, So let's just look at tables on here first, where we can get to keep us from by running it like best K. It's mild enough K plus dude. So now we can write on both sides because we know that the city this is food. We're assuming that the screw we get cable is on its more than f Okay, last three. Because we added want to the right side as all the left sense that we got three year now. Since we have three here, we can not just look at this one here. The statement. F gay Just three iss were recruits against improve F K plus one plus half Shea. Not a stake. My Beth. Let's still here. We can prove it. First told by expanding this, there's f K Plus three is smaller or equal to f. Okay, plus F K minus one plus two. No me know that the minimum number that the Fibonacci sequence can be here is one. And then this will be one to this will be to This is amazing one. That's the minimum right on. So we can say that it's quicker than recruiting for for every scenario, because F. K's here after is here. They're at it. F gate minus Lindsay Afghan minus one this year, which is, at minimum von this admin onto So the men on here with the three. So it's always either equality or queer than the scenario here. So since you have proven that now we can write it as K plus one. This Martin F K Plus three richest want Enrico to F K plus F K minus one was to which is equal to F K plus one plus two so we can be by this whole thing as K plus one. It's Morton F. K plus one plus two, which is very five.


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