WEBVTT
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here. PN represents a fifth fish population after and
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years. And that population is modelled by this formula
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over here. Okay, so for part A,
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we will show the following. If the sequence converges
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, then the limit it's and goes to infinity is
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either zero or be minus a. So let's go
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ahead and verify this fact solution. So let's just
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go ahead and supposed So we're supposing that it converges
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delicious. Call that limit l now using the given
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formula for piano up here. Let's take a limit
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on both sides of this equation as n goes to
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infinity. So we're applying the limit on both sides
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. So this becomes and this can be simplified now
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, too. The quadratic. Okay, so now
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at this that l equals zero is a solution to
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this equation over here. However, if l is
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not equal to zero, then we can divide it
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to obtain. And there are two solutions to this
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quadratic zero for B minus. And that was what
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we wanted to establish for parting. So if it
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converges the limit, it either dies off at zero
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or it stabilizes that B minus a, which is
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a constant. Okay with that said, Let's go
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on to the next page for a heartbeat. So
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this is where we like to show the following inequality
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. So it's quite and give a solution For this
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we have P n plus one, by definition or
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by the Riker JH in formula Given for p n
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. We can write this now. Let's divide top
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and bottom by a A number is just one,
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and then we get P n over a. Now
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I can just go ahead and ignore that denominator.
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The reason I can do this, we're dealing with
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positive numbers. Here's when the beginning A was a
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positive fish Population PM has to be not negative.
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So this is bigger than or equal to one or
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in this case, just equal to one. So
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that justifies this inequality here and that resolves the party
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because this is what we wanted PM plus one and
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then be over a PM And here this was You
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justify this. It's okay. Let me go on
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. I'll need some more room here, so let
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me go on to parse even on the next page
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. So here we like to use party to show
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that if a is bigger than be then Lim api
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and Ghost zero as n goes to infinity. Okay
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, so one way to show this is the following
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. So if the Siri's PN converges, then by
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the test for diversions we have the limited pn zero
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. So let's just go ahead and try to establish
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this fact here and then we'LL finish the problem.
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So let's try the ratio test for this using part
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see in mind because in part of seeing remember,
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she sees me from part B. We had key
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to the n plus one. It was less than
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be over, eh, Tien? So now we
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try the ratio test here. Since we're just dealing
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with positive numbers, we could drop the absolute value
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. This is B over a incense now and part
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see, we're assuming is larger than be that will
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imply be over in less than one so that the
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limit of Ki n plus one over p n is
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less than one. So that means that the Siri's
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converges by the ratio test and as we mentioned the
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beginning by the test for divergence, that will imply
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that the limit of P and zero and that's also
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what we what we wanted to establish a policy.
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Now there's one more party here, so we'LL go
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on to the next page for party. Now we're
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assuming is less than a B and there's a few
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things to show here. So first we want to
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show that if peanut is less than a B minus
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a, then we'd like to show that the sequence
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is increasing. And we'd like to also show that
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Pien is between zero on B minus eight Then on
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the other hand, if peanuts bigger than be minus
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a then we'd like to show these things The PM
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is decreasing and dance p and is larger than B
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minus a And then finally we'LL make a conclusion at
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the end that it is less than B and the
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limit of PN is b minus. Okay, so
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what's that? Improving this so and black thes two
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parts appear this worldview to parts and then two parts
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for the green as well. So this will take
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a few moments. So let's go ahead and start
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proving this. So we were getting so we're assuming
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this. But we were also given that peanut was
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bigger than zero, so we actually have zero less
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than Pena and then Liston d minus a. So
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now we would like to go ahead and let's show
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this one first. Let's do that by induction.
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We already have the bass case here. Bass case
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corresponding to an equal zero. Now, let's go
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ahead and uses for the induction. So we'd like
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to go ahead and suppose that it's true for PN
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and then show that would be induction. Okay,
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so now let's recall the following. We have tea
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and plus one by definition. Now, I'm just
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gonna do a trick up here in the numerator.
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And then I could rewrite this as the following.
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So now going on to the next page. So
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this was our assumption here for the inductive step at
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eight to both sides. Go ahead and divide.
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Bye, baby, he reckons, multiply both sides
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by the reciprocal zor football sides. And here let
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me put a negative motivated by what we had in
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the previous page. So this gives us again going
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back to the previous page here and then using our
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newest information. And that's exactly what we wanted.
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We wanted to show that if it was true for
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n that It was also true for en plus one
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and let's see here and by definition we know that
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zero will also be less than just by definition,
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of p m in the fact that a, B
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, t and R all positive. So this shows
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one of the parts we still have another parts ago
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. So now we have the following we now we
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want to show that it's increasing. So let's just
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recall that definition again. And then Now let's you
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not the family when we have what we already showed
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. So using the same type of trick is before
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and there we go. We see that PM is
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increasing sequence. So this takes care of the first
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part that we mentioned that was in black. And
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now similarly, there's two parts to show for the
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green steps. So this was if now we assume
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peanuts bigger than be minus a and so this is
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what we'd like to show. And then we'LL do
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it by induction again. Already we have the bass
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case up here. So now let's go ahead and
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recall the definition of P n plus one using our
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trick that we've used several times. So now since
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PM is bigger than be minus a. This would
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be our inductive step inductive hypothesis. It's at eight
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of both sides. Divide by eighty football sides and
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then multiply both sides by a negative on the next
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page. And so this will give us by previous
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work and then buy our latest inequalities. And once
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again, there we go. Now we have it
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for N plus one. Now we want to show
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that it's decreasing. So this was also supposed to
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be in green, but I moved on to a
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new page, so we'd like to show this.
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So we have add that aids of the other side
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using some algebra here. So this will give us
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that p n plus one bp on over a plus
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pn the shrink we wants than PM And so that's
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what it means to be decreasing. And so we
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had one more party here, and this one is
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to show that the following So if so, I'm
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getting a little sloppy. You're sorry about that.
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There was three parts. The party. That's why
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I'm using these down to you. So if peanuts
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less than a B minus in, then by what
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we showed this is positive, increasing and bounded above
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bye. Part by the first part of Heartbeat.
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We showed these things, and so his conversion bye
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theorems won't monotone sequence there and then also since the
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key and is bigger than zero and it's increasing.
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It can converse zero because it's increasing. So by
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partying we showed that the limit was either zero or
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B minus, and if it can be zero,
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it has to be B minus eight. Similarly,
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if peanut is bounded below, then buy this party
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. What we also showed is that pee and still
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positive. But this time it's decreasing. This is
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what we did in green and bounded below. Might
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be money, so it's converging also, by the
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twelve months a sequence there and then sense, um
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, Tien is bigger than be minus a. We
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have to have that the limit of P and bigger
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than equal city minus ing and which is bigger than
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zero so that the limit of PN is not equals
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zero and again by party. The limited it's non
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zero has to be a B minus, and that
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resolves the problem