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Numerade Educator

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Problem 21 Medium Difficulty

Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why.
$ a_n = 1 + (- \frac {1}{2})^n $

Answer

$1 \operatorname{since}$

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Video Transcript

the first few terms that we have here one half when we just plug in in equals one five over for when we're plugging in equals two and bunch of other terms. But it's more helpful. Tohave it written in decimal form zero point five one point two five Your point eight seven five one point o six two five zero point nine six eight Hey, one point o one five six zero point nine nine two two one point. Oh oh three nine your point nine nine eight Oh, one plant Oh, oh, no. Nine. Okay. And then Graff, these we label one through ten down here and then of our values get above one point two five. So we'LL just stop up here at one point two five. So for one, we have zero point five for two. We have one point two five for three. We have zero point eight seven five for four. We have one point o six two five for five for n equals five, we have zero point nine six eight eight for what sees we're on in equal six Now for an equal six, we have one point o one five six okay. And you, Khun, you can start to see that we're reaching some horizontal Assam towed here. All right. These dots are going to be getting closer and closer to this horizontal ask himto so it it does appear that we converge. Okay, it looks like we're converging toe one. And if we write the limit as n goes to infinity of a n, we see that, Yeah, we do end up getting one. Because as n goes to infinity, minus one half to the end is going to go to zero because one half is less than one an absolute value. So if you keep multiplying it by itself, then it's going to approach zero.