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(2 pts) The series X-1(3k2 3k)gives the total number of sticks needed t0 make letrahedral with Expand and evaluate the series t0 find out how many sticks are needed...

Question

(2 pts) The series X-1(3k2 3k)gives the total number of sticks needed t0 make letrahedral with Expand and evaluate the series t0 find out how many sticks are needed t0 make kite with 5 layers_(Ipt) Find the missing terms in the arithmetic sequence [5.(2 pts) Find the bth term ol the arithmetic sequence with dv = 64 and 012 = 88

(2 pts) The series X-1(3k2 3k)gives the total number of sticks needed t0 make letrahedral with Expand and evaluate the series t0 find out how many sticks are needed t0 make kite with 5 layers_ (Ipt) Find the missing terms in the arithmetic sequence [5. (2 pts) Find the bth term ol the arithmetic sequence with dv = 64 and 012 = 88



Answers

Evaluate each series to the given term.
$$
\frac{5}{2}+1-\frac{1}{2}-2-\ldots ;
$$

All right, so we want to find the sum of the 1st 6 terms of this geometric Siris. Now six is not that many. So the easiest way to do that is probably just to write out the 1st 6 terms. So what is the common ratio here? What are we multiplying by between each turn? Well, to me, it looks like negative five since two times negative. Five is negative 10 and then negative five again, right? Negative, 10 times negative. Five is positive. 50. So if that's true for multiplying by negative five each time, that means the next term would be 50 times negative five or negative to 50. And then that times negative five, which is 1250. And then that times negative five, which is negative. 6000 200 and 50. So if we add all of those together, it turns out that that is negative. 5000 200. Wait. But I want to remind you of another way that we could have done it, which would be to use this formula. But the sum of the first in terms of the geometric series is this. It's the first term minus the first term times the common ratio what to the 10th power, divided by one minus the commoner show. And if we were to do that on this series at six is thief first term, minus the first term times the common ratio to the sixth power and then divide by one minus the common ratio. If you were to work all of that out, it would also be negative 5208. Now the reason I mentioned this, even though it's kind of pointless here because we can just add up the 1st 6 terms you could imagine if they were asking for the 1st 100 terms, it would be completely invisible to add those together directly. And instead of being much more useful to use that formula where you can just plug in 100 for end and get the sum of the 1st 100 terms automatically. So anyway, that is how you find the son of the 1st 6 times, either directly or by using that formula

Here we are asked to evaluate the Siri's toothy 10th term. First thing we need to identify is what that 10th term will be. So let's get rid of this ellipsis here to find that 10th term on. And let's look at our pattern. It looks like from term to term were consistently adding two to the number. So if we follow that pattern, we will have 10 12 14 16 18 and then finally 20 Here we have 10 terms in total, and we could just add them all up if we really wanted to. But to really make this a lot easier and smoother, we can use this formula here. Uh, N is indicating the number of terms a sub Juan is going to be our first term in the Siri's. An ace up 10 is our final term in the series. So if we plug everything in, we can see that we've got 10 terms here, so the total sum will be our first term was too, and our last was 20. So two plus 20. If we evaluate what we have here, 10 over too is going to give us 52 plus 20 will give US 20 two, and then when we multiply these 22 times five, we get 110 as our final some of the series.

Okay. This question asked us complete the first five partial sums. This geometric sequence. Well, the first partial time is just the first term. So that will just be eight. Second partial sum is the sum of the first two terms. So eight plus negative four. That will just be full the third partial summers to some of the first three. Or we can just take the second partial sum and add the new term onto it. So in this case four plus two. The six. The fourth partial sum. Again, we can take the third partial sum and add the new term on here. So six plus negative one For five and the 5th partial sum. Yeah, we could add the first five up or we can take the 4th partial sun and add the next time on. So five, that's one half. We're all over and over two. So these are the first five partial sum. It's nice and straight.

Here we are asked to evaluate the Siri's toothy one hundred's term. So in order to do this, we are going to be using this formula here Capital s sub n and being the number of terms that we are going out to in this case 100. So let's make a note of that. So are, and it's gonna be 100. A sub one is gonna be our first term and a sub and is gonna be our one hundred's term in the sequence because our end is 100. Now, looking at this, the last term is not given. We don't know that hundreds term. Some people might be able to look at this and figure out what it is easily. But if you are struggling, you can use this formula here to figure out what any term is. So if we are trying to find out the hundreds term, we just need to follow this formula. A sub Juan is tthe e first term, which is to and then our end is going to be 100 still minus one and D here is the common difference. If we look back at our pattern and ask yourself. What's the common difference between all of these numbers? You'll notice that each time we are adding one, so are common. Difference is one from here We're just going to evaluate and what we end up with. First, let's follow our order operations and take care of what's inside the parentheses. We've got 99 then multiplied by that one on the outside will leave that as 99 then two plus 99 is going to give us 100 one. So our 1/100 term is going to be 101 referring back to our final formula that we use to evaluate the Siris capital s sub 100. Because we've got 100 terms, we've got 100 terms over too. Face of one is our first term. We've already discussed. It's too. Plus, our 1/100 term, which we just discovered, was 101. And once everything is Clifton, we simply need to evaluate using order of operations. This fraction out front 100 over to will be 50 two plus 101 is going to be 103 and then last bill, at least multiply 50 times 103 and you will find the final answer is 5000 150.


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