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Numerade Educator



Problem 82 Hard Difficulty

Show that the sequence defined by
$ a_1 = 2 $
$ a_{n + 1} = \frac {1}{3 - a_n} $
satisfies $ 0 < a_n \le 2 $ and is decreasing. Deduce that the sequence is convergent and find its limit.




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Video Transcript

for this problem. Let's go ahead and show this inequality first for all in, say, and bigger than or equal to one. Okay, so how should we show This lets his induction. So for induction, you have your bass case. This is when an equals one. Go ahead and plug that in. You have a one that's equal to two. This is given information, and this is a number that satisfies this inequality. So that's true. That takes care of the bass case. Now we go to the inductive step. This is where we suppose that it's true for some value for some, and now we want to show that it's still true even after you increase and buy one. That's the inductive step, and this is our inductive hypothesis right here. So let's go ahead and show that this is true. We have an plus one one over three minus a n that's given by this equation appeared the formula. Now this we can use the fact that so here, let's go to the side. So we have three minus a n. Let's look at this because that's in our denomination here. This is less than three because a N is bigger than zero, so it's positive number. On the other hand, it's bigger than or equal to one when you that's when you plug in a and equals two. So since this is in the denominator, this tells us that we have one third less than equal and less than or equal to one over one equals one, and therefore we have sees me here. There should be a plus one again because our denominators between one and three are numerator is on ly one. Some means the fraction as a whole has to be between a third and then one over one. So by induction we really have one third less than a n less than or equal to two. But let's really what will really need, and what we were really just asked to show was that this is true. This is implied because if Ann is bigger than the third and it's definitely bigger than zero and we'LL just take this week or staying there for now because that was asked over here. So that takes care of that first objective. Now our second goal. Let's go ahead and show that this isn't decreasing sequence so we'LL go to the next page. The next claim a N is decreasing, so it's monotone. Let's go ahead and show this also, by induction. We have our bass case. This is when we need to show A, too is less than or equal to anyone. So it's non increasing. So here we have a two equals one over three minus one one over three minus two is one that's less than two, which equals a one. So if you need here, they want strictly decreasing, and you could see here that it's not equal. It's just a strength in equality here. Now we go to the inductive stuff. So this is where we suppose that it's true for some value of and here we want to show that if you increase and buy one that it's still true. So increase and buy one. So both of these end values here on the left, you and plus two on the right, you get an A plus one. So there it is. It's a one down there, so this is what we want to show. So let's use the formula on AM plus two. Now this lets a denominator that's a n and an A plus one. After the end, this will be less than one over three, minus a m. Now let's justify that key inequality right here. So we know that this is true. That's my are inducted hypothesis normal supply both sides by negative at three to both sides. And we see here so again I shouldn't have included the equal sign. So these air shirt an apology is not equal to strict. So we see that three minus and a smaller So therefore that denominator smaller here. So the fractures the hole is larger and then using the formula on page one, this is a N plus one. So we showed that this is true. So by induction, a N is decreasing. Now we're almost done. One last step. We have to say, Why converges and then find the limit of the sequence. Let's go to the next page. So we've shown dead and satisfies this inequality. So this means that the sequence is bounded. That's important. Will need this. And we've also shown that a N is decreasing now. We consigned the room. Yeah, by the mama's home sequence. Their own monotone sequence. Carol, are the conditions are satisfied. The conditions are that the sequence is bounded and then it's Mama's Home. Doesn't have to be decreasing can be increasing but one or the other not both. So it's monotone and bounded. Data allows us to apply the theorem. We can say that the delimit convergence Let's go ahead and notify No and also observed that the limit of a and plus one is also equal to L. Because as n goes to infinity and plus one also goes to infinity. So the n plus one that will not change the limit out. So if we go ahead and take the recursive formula on page one, let's go ahead and take a limit on both sides as n goes to infinity. Now, this is equivalent to the left side. Is El the right side the numerator cz one, We have a three girls were Constance and then minus l none will supply that out and let's go to the next page here. So we just have to run Do elsewhere minus three l plus one equals zero. Go ahead and use the quadratic formula here or we have to be careful here. It could only have one limit, so we may have to cancel out one of these limits here. Some recall We showed this by induction earlier. This implies that the limit has to be also inside this interval, but it may hit in my record and point there. Also, it is wise here to include equal Sonny just in case. Therefore, we can only keep the answer. That's between zero and two, and therefore we could cross off this answer, and that's the limit of the sequence, and that's our final answer.