5

(20 points) Let fn(r) = e nI 3e-Snz . Show thatfn(c)dc #fn(r)dr . n=]...

Question

(20 points) Let fn(r) = e nI 3e-Snz . Show thatfn(c)dc #fn(r)dr . n=]

(20 points) Let fn(r) = e nI 3e-Snz . Show that fn(c)dc # fn(r)dr . n=]



Answers

Given: $D W=O N$
Prove: $D O=W N$

Show and just zero point coaching and juice and waste in the left hand side. We cultured and pictorial off reserve Toyo than a minute zero toil. And then we go to the infant trying always over time it could you want. Then we have the end tile and isn't exactly equal to the one now from the right hand side again in question and Troy over and editorial off and minus and from Toyo, you were getting acquitted in factorial and material doesn't leave is there of a toil it could you want if you're getting what you want, So he could, you know, let insight here.

Hey guys, In this problem, the textbook asks us to prove that n factorial is equal toe end times and minus one factorial. Now, in order to do this, we're going to be using the definition of the factorial, so let's get started. So first, let's start with the definition of an factorial. Well, n factorial is nothing but and times and minus one times and minus two times love, love, like all the terms in between, all the way to to times one. Now, another key note that we're gonna make on the side is we're gonna note the side the definition of n minus one factorial well and minus one factorial is the same thing as an minus. Uh, whoops and minus one. And mine is too Times all the way to two times one. Now take a look here. The definition of n minus one factorial is right here. So all the terms that comprise of n minus one factorial are included in and factorial so we could replace all of these terms with and minus one factorial. And in order to do that, we just substituted. So we say that this is equivalent end times and minus one factorial. And voila! We have the proof of the problem. We basically just showed that an factorial is equal to end times and minus one factorial Q e d or the end of the proof. Thanks for listening, guys. And I hope this helped, uh, show you guys how to understand this proof of concept that factorial is equal to end times and mine.

Okay, so we know by definition the backwards difference, uh, and is equal to and minus a and my ass one. Okay, but now we're looking for the relationship where we have a SMS one can alone on one side of this equality. And so how we're gonna do this, we can add Is that an minus one To both sides which will give us that the backwards difference, plus a set of M s one is equal to a n So what part were there? But no need to subtract the backwards difference from both sides if we subtract backwards difference from both sides of this equality we end up with is that a A minus one is equal to a n minus the backwards difference. And that is what we're looking for.

Yes. Well, here we want to show that a n minus two thirsty, backwards difference of a M. Woz thes second difference of a N is equal to a and minus two. Okay, so I'm gonna work with this left hand side here. It really just kind of get rid of rid of what we're aiming for and see if we reaching the end. And so we're gonna do is first just probably in the definition of the backwards and the second difference here. Right? So that would give us a N minus two times thieve backwards difference. A. And plus we'll get the backwards difference of a n minus D backwards difference of a and minus one. We get a cancellation, right? So one of these will cancel out with one of these the for the backwards difference. We're left with a n minus D backwards difference of a n minus the backwards difference of a and minus one. It's No one replaced these backwards differences with their their definitions. It was just what they are. And so this is gonna be in minus a and minus it and minus one minus a m s. One. I am minus two. Okay, well, sweets and cancellations. Right. So here we look positive. Am that counsels with this, This negative one of these two minus signs Here. So this ams one is positive, but it's gonna cancel out by this other subtracted a sub and minus woman. So we're left with again with the two minus signs. Hears we're just left with a positive A to the n minus two. Which is exactly what you We're trying to show just a little exercise in algebra here.


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