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Predlct tte naxt lira in the sequenc# of computations Then Use calcularot ~Usa Inductive reasoring Yar pelform the arthmetee handi determita whether your conjeciure...

Question

Predlct tte naxt lira in the sequenc# of computations Then Use calcularot ~Usa Inductive reasoring Yar pelform the arthmetee handi determita whether your conjeciure [ correc37472+344 52+3+45402 - 3+4-5Ane neu |t in Ine {equence 6 1+2-3.4+5-6 = (Dono stolty Ue? intcners Or Irucbons %0 any nunocts ehe @xpression | "n toneteure correci? Selatt Uie correcl answer below and ! Lhy unyil boliysi comoleecnoicoYes Dec4uso boan tne leh sice #13 the right side a2 eqqual toReemef Maa WAti Alt equaland

predlct tte naxt lira in the sequenc# of computations Then Use calcularot ~Usa Inductive reasoring Yar pelform the arthmetee handi determita whether your conjeciure [ correc 374 72+3 44 5 2+3+4 540 2 - 3+4-5 Ane neu |t in Ine {equence 6 1+2-3.4+5-6 = (Dono stolty Ue? intcners Or Irucbons %0 any nunocts ehe @xpression | "n toneteure correci? Selatt Uie correcl answer below and ! Lhy unyil boliysi comolee cnoico Yes Dec4uso boan tne leh sice #13 the right side a2 eqqual to Reemef Maa WAti Alt equal and Ine riatu s/d+ cquals



Answers

Let $T_{0}=2, T_{1}=3, T_{2}=6,$ and for $n \geq 3$
$$T_{n}=(n+4) T_{n-1}-4 n T_{n-2}+(4 n-8) T_{n-3}$$
The first few terms are
$$2,3,6,14,40,152,784,5168,40,576$$
Find, with proof, a formula for $T_{0} \mathrm{of}$ the form $T_{n}=A_{n}+B_{n}$ where $\left\{A_{n}\right\}$ and $\left\{B_{n}\right\}$ are well-known sequences.

You have to prove here. Eleanor C1 C two. So on cm. It's true for L. N. C. One plus L. N. C two. So on Ellen CN Mhm. For check base case first base case I learned C1 alleges will be a learner. Just see one and our ages will also be Eleanor just see you. So this case is proved now assume. Yeah. For induction step. Yeah assume first. For in truth learn off C one C two. So on CNN trooper. L. N. C one plus Helen Seattle so on 11 cm. Then to prove or and plus one which is Eleanor C one C two So on c. and plus one. This we can also write as a learner. C one C 2. So on CNN in two of c. n plus one. And we know the property of that log how many love him less. Low and this is the property we studying lottery thing so we can say love to make so 40 so we can ride the stairs Helen of C1 C two. So on CN plus a line of C. N. Plus one. And this term from the equation this equation one can be written as A line of C1 plus 11 of C2 plus stone, L n O c N Mhm Plus Eleanor, CN Plus one. So this is the same thing as RHS so this is just what we needed to prove. Hence, yeah. Equality is truth. Yeah.

Today we're verifying that formula or using the formula that the lack life transform of the integral of a function it was gonna be the lack last transform of the original function divided by S. We're gonna use this to calculate the last class transform off T to the end for all em on says we're proving this for infinitely many, uh, values. We're gonna be using induction. The induction begins the base case, which is that the LAPD transfer war of one is one over s and that's given to us. Um, and we're showing that it's equal to n factorial over esta bien plus one. Just to remind you, In which case, uh, this formula is certainly satisfied. Now, let's take the inductive case. We have, um, the integral from zero to tee off U to the N D u equal to t to the M plus one over and plus one that tells us that if we apply this formula, the lack last transform of tea to the M plus one over M plus one, it's gonna be equal to a lap last transform of tea to the end, which is n factorial over. As to the Empress one divided by s. Now we see this extra n plus one term, we can multiply it over to here on dhe, Combine these two F actors to obtain that the last last transformer of teacher, the M plus one. It's gonna be equal to end. Plus one factorial over s to the Empress. Dude completing the proof of the inductive case.

Okay, So you asked to use induction to prove that for all and Jill, each of the power of X is greater than or equal to one plus x plus 1/2 x squared. Plus stop at that was won over n factorial of extra bar Ben for excreted in her equal to Joe. Okay, so for induction, we start with an easy call to one day, If that's true. Well, that's this e t executed an equal to one plus x for excreted in Joe. And this is based on example or exercise 75. Right now we assume that this is true for an is equal to K soon. True. So that means yeah, you need to know it's good. In an equal to one plus x plus one of 1/2 x squared. I stopped at the thought. Just one older J factorial times extra virgin. Okay, now, if I take in a roll of this without X of the same thing Interval from zero x. Yes, but yet what in the middle of E. T. Is this Peter G evaluated at X and zero. That's this equal to each X minus one okay. And integral of the right hand side is equal to X plus, I mean these stuff. And one is this x Hardisty. Plus, she squared over two plus about a duck. T K bus went over one factorial evaluated at x zero. What does that give me? Well, we have E t X minus one is so this is greater than a it had been equal to. Let's inaugurate this at X Away yet X plus X squared over to put that under that plus extra. Okay, plus one all over K plus one factorial minus that. Evaluate a girl with a field goal. Joe, we don't get anything about that now. Let's move. Let's add one to both sides. I would eat it, executed an equal to one plus x plus X squared over two. Let's put it on. That was extra power Cave us one all over K a close one factorial. And this is actually And if you go to K plus one since this works for K plus one, my induction, this is true or all and greater than two good it in her equal

Okay. First I want to say that in the problem. I think there's a typing mistake. I think on the very last one. The main one over in factorial X to the end because two is two factorial Sixes three factorial, 24 is four factorial. So I really think they meant in factorial there. Okay. So we're gonna do the proof by induction. So by induction, the first thing you have to show is that E to the X is greater than or equal to one. Ah For all X greater than or equal to zero. Okay, but we we showed that already. Well, we'll do it again. Um He is greater than or equal to one and E to the X is greater than or equal to E. So to the X is greater than or equal. Don't want. Okay. Second assume the statement. It's true for in equals K. That is a cell. Yeah. Eat to the X. Is greater than ankle one plus X plus X squared over two factorial plus executed over three factorial plus dot dot dot plus X to the K over K factorial. So, we're assuming that we know that that's true. Okay. Step three. Show the statement is true for n equals K plus one that is show each of the X is greater than reclaim one plus X plus X squared over two factorial plus execute over three factorial plus dot dot dot plus X to the K over K factorial plus X to the k plus one over K plus one factorial canal. If we can show that then we have shown by induction that that statement is true For all in greater than or equal to zero. All right, well, it's easy. We'll just start with since the thing that we know it to the X is greater than or equal to one plus X plus X squared over two plus dot dot dot plus X to the K over K. Factorial. Then By theorem one in the last problem. 02 X. E. To the T. D. T. Greater than or equal to the integral one plus X plus. Oh dear to X. One plus T plus one half a T squared Plus 1 60 cubed plus dot dot dot plus one over K. factorial T to the K. D. T. The young girl on this side is key to the T from 0 to X. And so that is greater than wrinkle to the integral over here. Which is T plus one half a T squared plus one half times T cubed over three plus 1/6. T. To the fourth over four plus dot dot dot plus one over K factorial T. To the K plus one over K plus one From 0 to X. So each the x minus each of the zero which is one is greater than T plus one half T squared plus one. What's 1 60 cubed Plus 1/24 T to the 4th plus dot dot dot plus teach the K plus one. Okay, if you multiply K factorial times K plus one, that is K plus one factorial. So each of the X is greater than Wrinkle Awan can. Now I'm adding one to both sides. T plus one half a t squared plus dot dot dot. Let's teach the K plus one over K plus one factorial. Okay, that's what I was trying to prove. And so by induction, each of the X greater than Wrinkle one plus T plus one half a t squared Plus 1 60, cute plus one over in factorial, teach to the end.


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