find-the-nth-term-of-the-given-infinite-series-for-which-n123-dots-frac12frac14frac18frac116dots

Question

Find the nth term of the given infinite series for which $n=1,2,3, \dots$
$$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\dots$$

Tyler Moulton · Accepted Answer

Yeah. We want to identify the term of the infinite series for which an equals 12 and three. And our series is given as one half plus 1/4 plus 1/8 plus 1 16 and so on. This question. And challenge your understanding of sequences and series in particular is challenging us to be able to identify the pattern which constitutes the given infinite series of sequence. So let&#x27;s first separated are terms that will make it easier to identify the pattern present in this series. We have a one is one half to two negative 1st 82 is 1/4 19 2nd. A three is too negative 3rd and 84 is too negative fourth. This is like manipulating one half, 1/4 1 81 16 appropriately. So another one identify the pattern going on. We see that each of a one through four is to racism power. Where that power is the negative of the sequence number. So for a one or two to the negative 1st 80 to 92nd. So on the pattern we see A. M. Is too negative then, which means that we can identify A. M. Equals one over to to the yet.

Subhadeepta Sahoo · Answer

even see this would consider to be 1st 81 in the common minus of the N F. Slash of some and one No, there isn&#x27;t one of one minus minus one, despite one less going to find some of the city&#x27;s. So what? Limit? So that score to the dentist group three right to bend one minus. So when interest to infinity so I&#x27;m certainly pretty could hurt the country.

Subhadeepta Sahoo · Answer

When the defensive is written, see that there is an energy be extracting. May 2 41. Come on the line since, huh More is great content. One. So we concerned that the Syrians that&#x27;s not convert antibiotics, but the next part through this in the quarter, booking a pardon one minus one three. I didn&#x27;t have much my group.

Katelyn Chen · Answer

you&#x27;re given a follow Siri&#x27;s and pacifying the similar Siri&#x27;s. The first thing you do is you can factor out the four to get one over for our own, my three times for one plus one, as the next thing that you need to do is you need to apart a telescope. Serious fast stuff you have won over four and 1st 3 over four plus one is equal to a over or earned my street plus three over four n plus one. So if you lost by four and minus three and foreign possible into both sides, you get one is equal to hey, times four and plus one plus B times four on my three. So if you said and let&#x27;s just i n is equal to native 1/4 that will make a equal to zero, because four times and No 1/4 plus one is just equal to zero. And for B, you will get so you get beef is equal to negative for okay, so B is equal to one divided by night before, which is not 1/4 you need to saying applies in concept to be so you said and is equal to 3/4 and that sets beat zero. So you just get one is equal to and then four times three forced plus one is simply for hay, so a is equal to 1/4. So give us information because simply plug it right back in. So you have this function, and you can set it equal to four. So one where you know you now know it. So you know, that is 1/4 over four in life. Three Linus, 1/4 over for Aaron. Plus one, actually, can you just simply factor out the 1/4 it&#x27;s that will get you 14 times for just one? So get the Siris of 1/4 and my story minus one over for an plus one. So to find a some of Siri&#x27;s, you can simply plug in different terms to find a pattern to find the formula. So you comply. That s test to find the formula for the Siri&#x27;s. So you will get your some of your serious equal to one minus 1/5 waas. One fish, Myers. One night, plus more knife minus 1/13. Sandra plus 1/4 on my three minus one over for what? And she is so much are canceling these out. So, you know, one of my politics. Zero the ninth wise one. I just arrow etcetera and you are left with one minus one over for n plus one. And you know that as on approaches, infinity, who are impossible is just to get larger and larger and larger. So it is going to approach Sarah. So you know that Siri&#x27;s as under purchase and Fendi will approach one. So one is your answer.

Kevin Shryock · Answer

Hello, everybody. And welcome back. This is Kevin Truck with New Murad. So we&#x27;re given this infinite Siri&#x27;s. We have six divided by the quantity two and minus one times the quantity to n plus one all in the denominator. And as you all know, what I like to do is to just take a moment and expand this out a little bit and see how this syriza&#x27;s behaving. So we have six. And then if I look at that first term, it&#x27;s going to be two times one minus one. So that&#x27;s going to be value of one times two times one plus one. So it&#x27;s gonna be a value of three, and that becomes the first part of my serious. And I could do this for the next value as well. So it&#x27;s consider six, and then I&#x27;m gonna put in a value of two. So it&#x27;s gonna be four minus one is three and the next value is going to be two times two. So four plus one is five and we continue this process on and on and on until we get to the partial summation. If we decide to end it at some place, so we could. We could kind of interrupt this for a moment. It will say the term would be six over two times, Um, minus one times two m plus one. And we could stop it there or we could continue it going. Now, after playing around with this for a little bit, there&#x27;s not gonna be a pattern that is immediately obvious here. So what this is gonna make us do is just turn to branch out. This is definitely not geometric. This is not something we have. Ah, nice, immediate formula for So instead, what I decided to consider was maybe doing a partial fraction decomposition. So enough, go ahead and open up a new page here where I&#x27;ve got it copied again. And let&#x27;s consider what this partial for action decomposition might do it. Sometimes that reveals that we can get sort of like an alternating opposite kind of components sequence. Let&#x27;s let&#x27;s let&#x27;s do some exploration. I&#x27;m going toe factor the six out front here, though, because that&#x27;s gonna save me a little bit of a little bit of trouble. So I&#x27;m gonna go ahead and get rid of this six. I&#x27;m just gonna make us into one. It&#x27;s gonna make my math a little bit. He is year. So we&#x27;re gonna do a side note. So consider this side note. Um and I&#x27;m going to say, What if I had 1/2 and minus one times two n plus one and I want to decompose this into some value of A that is over two and minus one, plus some value of B over two and plus one. What we can do. A little bit of simplification here that&#x27;s going to show us that one is equal Teoh to and plus one times a plus two and minus one times be just like this only does some common denominators and did a little bit of solving on through Well, from here. Let&#x27;s continue to solve. So we&#x27;ve got one is equal to two and capital a close. Hey, plus two n Capital B minus B. Now notice. On the left hand side, there are no end values, but there are some some unit values, this value of one so we can kind of do a little bit of grouping here and start to make a system of equations and recognize that we have two equations here. One is equal. Teoh a minus bi. And then we also have these n values that don&#x27;t appear anywhere. Even think of this as being like a zero n is over here. Well, then we can write that as being zero is equal to two A plus to be some quick substitution gives us that we have B is equal to negative A and substituting that in over here gives us a minus and negative A is equal to one So in total were able to conclude that a Then it has to be 1/2. If a is 1/2 then be must be negative 1/2. Okay, so we&#x27;ve got we&#x27;ve got our full breakdown here that we can now go back to our original process may be redefined this a little bit nicer. So let&#x27;s say this equals six times the partial summation from an equals one to infinity Uh, 1/2 over two and minus one. So I&#x27;m just going to certify that the 1/2 to an a minus one and then plus negative 1/2. So it&#x27;s I should just make that a minus. We&#x27;ll just make that minus 1/2 two and plus one. And in case I lost you there, all I did was I took these A values and the values that we&#x27;re over here. I re combined them with the original structure that I was trying to replace it with and then substituted on it. OK, so now let&#x27;s look at the first few sequences 1st 2 values of our Siri&#x27;s. So you get six times and then let&#x27;s begin to expand this out when and is equal to one. This is going to be 1/2 times. Two times one minus one. Okay, so we&#x27;ve got two times. One is two minus one is once. We&#x27;ve got 1/2 of the first term, 1/2 minus the first part of the first term. One over balls just to two times. One is two plus one is three times two is six. So that&#x27;s in the 1/6. And that&#x27;s the first term in our serious Let&#x27;s do another one. They are putting an end value of two, so it&#x27;s going to be two times two is four minus. One is 3/10 to its six. This is going to be 1/6 and then we&#x27;re gonna have two times two is four plus one is 52 times five is 10. So 1/10 minus 1/10. If you continue this all the way through even if you get to the value, you&#x27;re going to notice that there is a process that is repeating itself. So I&#x27;m just gonna simplify this up a little bit. We have, um, for em for our end term for em, minus one or minus two. Excuse me. Minus one over for em. Plus two. Anson, if installation. So that continues even past the end of term. But what we noticed is that if I was to stop this at any given moment, so maybe I redefined this line of work and I call this my partial, some upto em. But if I can labels of my partial something to take away everything after em and look at how those terms behave. What? We noticed that there&#x27;s this opposite canceling pattern. We&#x27;re pursuing the beginning. So this negative 16 is gonna cancel out with this 16 This negative 1/10 is gonna cancel out with the first term of the next part here. And so we continue cancelling it on through where these is later terms cancelled within earlier terms. That means this one over Forehand minus two is going to be gone, and I&#x27;m left with this. This last term is my only little bit. So we&#x27;ll see if we can squeeze in some room here at the top. This kind of section off a little bit space right here and see if we can wrap this on up. So that means that our partial summation up Teoh em is equal to 1/2. Let&#x27;s not forget about tag along 66 outfront six times, 1/2 minus one over for em, plus two. And we can consider what would happen if I just kept extending out this this value of them. So let&#x27;s look at the limit as M goes to infinity and we could notice that this is all going to converge. The ESA bam is going to converge to the second term, will drop away. And so I will be left with six times 1/2 or just simply

Vishal Parmar · Answer

Yes, we&#x27;re in this question. We want to find the first some of the 1st 20 time off this arithmetic sequence. So here n is equal to three bedroom times in must read. Right? So, first of what would an easy quarto one. So here it once 1/4 to 3 bite, sometimes one money time. I will. Right? So here are really multiplied by two that does he go to 24 and 24 plus three. That is equal 27. So here went descend by two Easy. There were flushed. Um Right now, take an easy quarter. 20. So a 20 is equal to three by through dance trending last trail. You&#x27;re 20. Divided by true that is 10 and then multiply by three That is 30 and 30. Plaster that is equal to 42 20. Tommy&#x27;s 42 the forced Thomas 27. I do now some of the arithmetic sequence that is the thinnest Eun bi And by group games for stump Plus last time, Right? You&#x27;re an easy Gordo. 20. So, yes, 20 is a good 1 20 divided by two. You&#x27;re a born. That is a Bordeaux 27 by two in the last time, that is 20 is equal to 42. 20 by Luis. 10. Right? You&#x27;re 42 months led by two that is equal to 84 84 plus 27. That is a good 3.1 you wanted by two. Now 10 divided by through that is equal to five. So here Ansari&#x27;s people one. Do you read by my So you don&#x27;t do what? A bat flying about Mont education My five right and triple one multiplied by five. That is a world tripper five. That is the sum of the first Grandy terms. Thank you.

Problem

Find the nth term of the given infinite series fo…

Problem 12 Easy Difficulty

Find the nth term of the given infinite series for which $n=1,2,3, \dots$
$$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\dots$$

Answer

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