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Problem 64 Hard Difficulty
(a) Determine whether the sequence defined as follows is convergent or divergent:
$ a_1 = 1 $ $ a_{n + 1} = 4 - a_n $ for $ n \ge 1 $
(b) What happens if the first term is $ a_1 = 2 $ ?
Answer
(a) Sequence does not converge.
(b) This sequence does converge (to the value 2)
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Top Calculus 2 / BC Educators
Catherine R.
Missouri State University
Heather Z.
Oregon State University
Caleb E.
Baylor University
Kristen K.
University of Michigan - Ann Arbor
Watch More Solved Questions in Chapter 11
Video Transcript
whose problem it helps to just write out the first few terms. So we know a one is one Hey, two is going to be four minus a one. So four minus one, which is three pay three is going to be four minus a too. So four minus three, which is going to be one. And from here, we know that the sequence should just continue in this way. One, three, one. But, you know, if you really want to convince yourself, you keep going. I thought I was going to be safe. Four minus a three is four minus one, which is three. A five is four minus a four four minus three, which is one. So this just switches between one and three, so that's definitely going to diverge. Okay. And once we got Tio Tio here, we knew that it was just going to switch between one and three, because each term is just defined only by the previous term. Okay, once you've gotten to hear, we've already been in this situation once before, right? So we know that if one is the previous term, that next term has to be three. So once we get to this one. We know that the next term has to be three. Okay, And then once we're at three, we know that the next term has to be one, because again, each term is defined only by the previous terms. So if we've been in the situation where we've gotten one before and the next term was three, we know that if one shows up again, the next time has to be three. And likewise, if three shows up in the next term is one, if that ever happens again and the three shows up, the next term is going to be one. But it's always good to, you know, you really convince yourself of it by, you know, right, Not a few more terms and seeing that indeed it is just going to alternate between one and three. So certainly divergence
Top Calculus 2 / BC Educators
Catherine R.
Missouri State University
Heather Z.
Oregon State University
Caleb E.
Baylor University
Kristen K.
University of Michigan - Ann Arbor