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(Kn)_ show that is AH9. (Divisibility:) Let n € N In this problem aTe going integer for all k € N: Show the k case_ Hint: Binomial coefficient?) Frove t...

Question

(Kn)_ show that is AH9. (Divisibility:) Let n € N In this problem aTe going integer for all k € N: Show the k case_ Hint: Binomial coefficient?) Frove the_general Gase (Hint: Try induction (d 360 Qour deriation:vou Caw fund Ci"t"10. (Elementary Number Theory:) Show thau for aIl /eN "2 diviclos Hint: Usc Binomial Erpansion to eqmd (" + and argue (hat n2 divides ech tcrm

(Kn)_ show that is AH 9. (Divisibility:) Let n € N In this problem aTe going integer for all k € N: Show the k case_ Hint: Binomial coefficient?) Frove the_general Gase (Hint: Try induction (d 360 Qour deriation: vou Caw fund Ci"t" 10. (Elementary Number Theory:) Show thau for aIl /eN "2 diviclos Hint: Usc Binomial Erpansion to eqmd (" + and argue (hat n2 divides ech tcrm



Answers

Show that $9^{n+1}-8 n-9$ is divisible by 64 , whenever $n$ is a positive integer.

In this problem of mathematical induction we have to prove. Given a statement using principle of mathematical induction for all and belongs to natural number. First we consider given a statement as the offense. And given his statement is 3 to the power to in plus 2 -18 -9 is divisible by mhm eight. Mhm. First we prove for three of one thing N equal to one. So we have 3 to the power 4 -8 -9. Simplifying this. We have This will be equal to eight even And this is 17. After subtracting we get 64 and 64 is divisible by eight. So it is true for the given a statement. Now we consider P off Cape. It will be three to the power to K plus two minus eight K minus nine. We consider this statement is multiple of eight. So 8 m. here I am equal to any national number. Mhm. Now we need to prove a statement we of K plus one. So we put K plus one in place of. And so it should be Yeah three to the power. Two K plus two plus two minus eight. Multiply by K plus one minus nine. Now we simplify the expression so it should be 3 to the power to give plus two multiplying three square minus eight K minus eight. And this is minus nine. Now we can write these three square is nine, Multiplying 3 to the power. Okay, last two. This is -8 eight K. And this is -17. No we arrange it like we can find the of K. So we write nine multiply by three to the power to K plus two minus nine multiply eight K -9, multiplied by nine. And also nine Multiply by eight K plus nine multiplied by nine. And this a k minus 17. Now we're taking common from this time this time and this time we have value equal to nine, multiplied by play to the power. Okay last two -8 K -9 and plus nine multiplied by eight K. My lip plus 81 -8. Game -17. Now we can write this whole time is 18. So it will be nine multiplied by eight. a.m. And also mhm. We arrange this to them Taking common so it will be clear to nine eight k plus nine and this time -8 K -17. Now Mhm. Multiplying this two times we have value equal to nine multiply 18 plus. So they okay plus 81 -8 K -70. We simplify the expression. So we have nine multiplied by eight a.m. Plus 72 K minus eight K. It will be 64 K 81 minus 17. It will be 64. Now we can see we can take common egg from. Also we have nine mm plus eight K plus eight. We consider this whole time sq so Q is an initial number. Mhm. So we have those days. Prof K plus one is multiple. Love eight means it is usable by eight. So we can say given a state May is to for all and belongs to national number. And this will be our final answer. Mhm.

High in this question. We are to prove that the number it divide the term kids choir My last one whenever K is our non negative vintages and we asked to use induction. So let's do this. This is one of those statements that can be proven on this own without using induction, I believe is by proving without induction is simpler, is what I believe. But they asked us to use induction. So Okay, now let this Steadman bp up in where in here is not the same ass. So this I used care, right? The term square here. But it is what made up the old number. So Kay is written as two in plus one. And since K is non negative what numbers and who run from 012? I do this just so that we have, like, the usual template that I use. You can index it in other ways as well. So here it is mean that we want to prove that is true fall for in where Indies these list, right? And so basic step. When Ennis zero, it means Ed Divi One man's wanted zero So and device your wishes. Obviously true And so next the inductive step He supports us that Ben is true for some in and we want to show that is true fall for in Paswan Mass Bill. But here again, I I want to know that I want to say that we will use information that we already know is true But it's not coming from p o in. But that's still like the proof is there? Well, it as long as you do. The fact that you're using to pro pee in 1st 1 is like, true on his own. And we are going to do that. So how do I show P off in plus one? Well, first, look at the term here. This is the term in this statement O p n plus one. Ah, because when we start, we have Eddie White kids choir minutes one white and the K we use East two in plus one. I mean, that's the P o. Instatement then for foreign parts. One statement we have to act like chin into in plus one. And so we had to to the K All right now, once we have this eye does factor, I multiply it true and then factor it into this too compliment and you sing. You're seeing this interpretation or this formula for K in terms of in I share this too. This too factors. And now you see that that is, there are number twos in both of them. So I pulled him out. Become this number four. What's left is in plus two ton in plus one. Remember, our goal is to show that in divide this holding right and we already have full. So what's left is if he can show somehow that to divide this this pod, like not not concerning for to divide this part than Ed would divide a holding using full here and and to from here, right. And how do we show that well is quite obvious because and plus one, an impulse to, ah, two consecutive images. All right, there's nothing going on that in 1st 1 you add one more. So is in plus two. When is two consecutive vintages one off them have to be Even so either this is even. Oh, this is even like one of them has to be even right? I hope this convince you. And so if that's if that's the case then too. Must be Why one off them? The one that is even So, I have been op tech Deco issue, OK? And so and so we have that to divide this last pod. And as I said, that means Ed who divide the holding because we have already have number four in front and to divide what's left. All right, I hope this convince you that that the state men fall and plus one is two, all right? And we don't exactly juice information from P off him. But whatever we use here is already true on this own. So it doesn't. This doesn't violate the proof by induction. All right. So, back to induction The inactive step is true. And so we have prevent board steps. So the statement is true. Fall for in which is to say is true for all on positive, often on negative integers Kay, which is what we want. That is it. Thank you

Hey, it's Claire is suing you right here. So if an and care positive and teachers than an a plus one times I choose T minus one over K is equal to an plus one tons and factorial over came minus one factorial I was on plus one minus cane factorial all over K. This is equivalent to and plus one times on Factorial overcame Linus one factorial comes and plus one minus King Factorial all over K times K over K. I know that these two were gonna cross out, so we get equivalent to a close one times and factorial all over case factorial times and plus one minus. Chief Factorial equivalent to and plus one factorial over K Factorial comes and plus one minus K factorial. Is this equal to bond plus one choose K.


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