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Calculate the sum of the seriesa, whose partial sums are given by s = 2 - 3(0.8)"...

Question

Calculate the sum of the seriesa, whose partial sums are given by s = 2 - 3(0.8)"

Calculate the sum of the series a, whose partial sums are given by s = 2 - 3(0.8)"



Answers

Find the indicated partial sum of each arithmetic series. The first 21 terms of $2+5+8+11+\cdots$

So with this problem, we're going to be finding the some of the Given Siris in the syriza's two plus four plus eight plus dot, dot dot in the last time that we were going to be adding is 128. So well, this man look like an arithmetic. Siri's, we're actually doing with the geometric Siri's cause two times two is 44 times two is going to be hate. So we're going to be multiplying by two every single time. So in order to find this well, we know that this 1 20 we're just gonna continue the Siri's So two plus four plus a plus eight times two is 16 16 times two is 32 32 times two is maybe 64. 64 times 64 is equal to 128. And so after you figure out this, you're going to just add all of these problems together with your calculator. And so we have two plus for a 66 plus eight is 14 14 plus 16 is going to be 30 plus 32. It's going to be 62 and plus 64 is going to be 126 and plus 128. We're going to get an answer of 254 and that is how you solve that equation.

Okay, so we need to find the sum of all these using this form of here. So it's counter terms. 123456 Uh, just like the previous from I did for this six over too. I'm just fucking in six for n. Okay. Two multiplied by the first term is zero. I just kind of interesting, but his 2 10 00 this is actually gonna go away Eventually. I need to add six minus one. And you know what? Let's say some plan. Six months, one is five. And what is the common difference? Well, for I for I for i for I cz added every single time. So let's go ahead into this, uh, six number two's three. Like I said before, two times zero is zero. So two times zero eyes also zero five times four is 20 times. I is still 20 I. So I get a three times 20 I, which is a 60 uh, is mice

Okay, so we want to find our 21st partial sums that it's us of 21. What X is equal to 21/2 times our first term. That's given to be negative, too, and then we want to add or a 21st term. So in order to find this, let's start by finding an We're not equal to our first time plus and minus one times I call the difference. So we take the difference of our 1st 2 consecutive terms. When we get 10 I know we'll use this equations he saw for a 21. They're supporting this in super tacky. There we get 198. So let's click. Listen to you. So let's now put that squeeze it into our calculated defines our, um, our 21st partial sound. So we see that we get, um, 1058

Let's go ahead and calculate the sum of this Siri's who's partial sums air given by the following formula over here. So let's distinguish between the N and the Essen. So recall by definition, this which is the and partial some that's our notation. That ass notation is further the partial. Some and the end is telling you how many terms or in the sun and this is the fines and be a one. Let's write it this way. The sun from Let's go k equals one to end of a K. So this is just the sum of the A's. But on Liam to some a m. So now and our problem there telling us what s it is they're saying if you add the first and numbers, you get to minus three, zero point eight to the end and now we want the infinite something so here and equals one to infinity of a M. That's equals the limit. Let's write it this way, since we're there using end here, let's write it as Argos to infinity and equals one. So are of I am Caesar that this right here. This is just as our so lim as our goes to infinity of s are so that's two minus three point A to the O. R. And his fueling our goto infinity and take the limit. This term right here goes to zero because point eight. So the R goes to zero as our goes to infinity and the way to see that his point is between zero and one. So each time you multiply by point, you're getting smaller and smaller. And so here this will go to zero and we're just left with two, and that's our final answer.


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