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1. [u 'pts] (Induction with Fibonacci) Recall the Fibwonacci seqpuence given by the recurrence F- for 2 2 Along with Initinl tetmus F 0 6=L Une mathetnaticul i...

Question

1. [u 'pts] (Induction with Fibonacci) Recall the Fibwonacci seqpuence given by the recurrence F- for 2 2 Along with Initinl tetmus F 0 6=L Une mathetnaticul induction to prOVA tlnt for " 2 [FF+ R+()

1. [u 'pts] (Induction with Fibonacci) Recall the Fibwonacci seqpuence given by the recurrence F- for 2 2 Along with Initinl tetmus F 0 6=L Une mathetnaticul induction to prOVA tlnt for " 2 [ FF+ R+()



Answers

Fibonacci Sequence $F_{n}$ denotes the nth term of the Fibonacci sequence discussed in Section $12.1 .$ Use mathematical induction to prove the statement. $$ F_{1}^{2}+F_{2}^{2}+F_{3}^{2}+\cdots+F_{n}^{2}=F_{n} F_{n+1} $$

In the Fibonacci sequence, you're given a formula that the reference was F two plus F three. So a fan is equal to have and plus two minus Martin bs. Approve it using induction. Their end belong study it and national numbers. I just ate our base case and is equal to one. Now we know that's one. So we get one. Is he closer? F three minus one, bridges three minus not three to minus one. This is equal to one. This is proven or baseless is right now foreign national policies taken physical decay since I'm physical. Okay, we get something like F one plus f teeth plus f k is equal to f K plus two minus one B after. Assume that this is true we use is there. So let's take invisible to K plus one as our induction case. Be active proof this is right. So we'll do it something like this. Someone like this at one. Plus f terry less f k plus f Okay, plus fun is equal. Teoh f Okay. Plus three minus one. Using more industry hypotheses, we can separate this part and this part. We can rewrite this as something like F K plus two minus one plus f f K plus one is equal to have K plus three minus one. We have to prove this right. According to the Fibonacci sequence, a number there is, um, succeeding Another number is ik with some off the number that's receiving yet and the some of the numbers preceding the other. The number that is proceeding the other numbers so they can see that f k plus one. And if keep was to directly proceed each other on and f capers deal darkie Proceeds of capers tweet. So this was this is equal to this. So we can be workers. I left Cape Leslie minus one is equal to f K minus one. So let's just go toe Rhs.

Given the statement. F warm clothes F three F two and minus one is equal to the F. T. When we have to prove it for n belong into and I just take our base cation is equal to one force now and is equal to one. We didn't find out later we can find out the basic issues in half. One is equal to have two into bridges. F one is equal deaf to just one is equal. Divine answer for it I was just looking at our induction hypotheses that is N is equal to K. We have to museum that an musicals case is correct. Our is full. This fruit F three plus F to K minus one is equal to F to Kate. No, since this is food we now have to prove and musical two K plus one That is our election case. So I just We write this as F plus F one plus F three F to K minus one plus F to K plus to keep this one minus one, which is equal to F two K plus two so we can separate all stem of into two parts using using the induction hypotheses that made are here on. We'll get something along lines off F two K plus F two k plus one is equal to F two k plus to me. No using the Fibonacci sequence that a term is the sum off its for seeing in terms of the term proceeding excruciating term. So have two cables on enough to counter two terms. Lucienne, that's to give us too. So f two que les two is equal to F two. Okay, let's to

High in this question. We are again talking about Fibonacci sequins and we asked to show that this formula here is true for any positive integer in. So I will let them be the statement B p O n and yeah, I know that the sum up to like and and I with two in always. So we have to add unto we hit you in terms. All right, that's gonna be important Later. Now we will use induction on this on in first Basic step or best them and equal to two. The statement said that if zero times if one so is sewer times one plus if one times if two wishes one times one equal if two square So is if one square. And this is true, right, because it's one equal to one. So our basic step is clear now for the inductive step. We suppose the statement is true for some in on we want to shoulder is also true for P and plus one as Bill. How do we do that we can stop with people in We will have this equation as true from the assumption then I add this to terms two new turn to the equation A ball. So the right hands, I will be the Squire plus days too. And notice that the sum on the left hands I now become the some indeed in the in plus one statement that the one what's left is to rearrange the right hands. I write first. This two first 22 becomes this by just pulling f to an out and inside like what's left inside gonna be if two in pass it to impress one, it become have to impress too. All right, then we have this with the last term again. We pool F two in plus two out and what's left inside will add up to have to impress to again. All right, just is just using definition off you. But actually nothing complicated here and we're gonna get if to impress too square, which is so this is the Steadman P and pass one that the one right some up to if to end us to get f two in pursue Squire. So we have shown the inductive step is true. And once we show in both states, this statement is true for any positive interjection and yeah, sorry, I I ah is not too Is one okay, so in equal to one. But this this form here is if up to in square so is basically have to square. Okay. I am so sorry. Please don't be confused. So the formula started in the quarter one, but it talks about ever to it. All right, that is it. Thank you.

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


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