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12. Prove that f}+ J+ f+_.+f= fnfotl using weak induction. (Hint: These are the Fibonacci numbers: f = f-1 + f-2)...

Question

12. Prove that f}+ J+ f+_.+f= fnfotl using weak induction. (Hint: These are the Fibonacci numbers: f = f-1 + f-2)

12. Prove that f}+ J+ f+_.+f= fnfotl using weak induction. (Hint: These are the Fibonacci numbers: f = f-1 + f-2)



Answers

$f_{n}$ is the $n$ th Fibonacci number.
Prove that $f_{1}+f_{3}+\cdots+f_{2 n-1}=f_{2 n}$ when $n$ is a positive integer.

Okay, so we're problem. 12. Uh, we need to prove that is by induction. The pieces stab we can see there is when he's equal to one, we have at one time said to which is equal to one times Juan. Is this the same as Apple is clear now, when that's assume that went and hit was to Kay Now where any was quick. Sorry, Wendy. When n equals k. Well, you have everyone squared. This is our assumption. Uh, plus have k squared, which is people too. Uh, okay. Times f k. That's one. Okay, now, now we want to show that when and then is equal to Okay, so we're problem. 12. Uh, we need to prove that is by induction. Oh, the pieces step we consider is when these equal to one, we have a one time step two, which is equal to one times one. Is this the same as happens? Clear now when that's assume that went and hit was to Kay. Keep us one. The what? The quality steel holds Now on the left inside. And we have one squared. Plus, I got the top plus f k squared. Plus. Okay, that's one squared and we apply our assumption here when he was K So some off this part will be a times f k less one now when he was quick. Sorry, Whitney, When n equals k Well, you have everyone squared. This is our assumption. Uh, plus have k squared, which is equal to f k times f k. This one. Okay, now, now we want to show that when and when then is equal to plus que plus quand squared. Now we take f k plus one. Uh, yeah, we take out we take out f k plus one. So we have kay. That's one outside times f k class F K plus one inside. Now notice that after a plus that keep us one is equal to F K plus two. So this is equal to K plus one time. Keep us one. The thief. The quality steel holds Now on the left. Inside we have one squared. Plus, I got the job plus f k squared. Plus Okay, that's one squared. And we apply our assumption here when he was k. So some off this part will be f k times f k less one. Okay, that's too So we're done because, you know, this is exactly what we need to show, which is that one plus f k plus want I appoint squared plus until, uh, f k plus one squared, which is equal to a Capel's one time second papers, too, so that

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, 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.

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


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