Find the sum of the first terms of the geometric series | + +( the least value of n such that the sum exceeds 20.Hence, find(5 marks)...

For each sum, find the number of terms, the first term, and the last term. Then evaluate the series.
$$
\sum_{n=5}^{10}(20-n)
$$

All right, So this is a question on a geometric Siri's and, uh, the Siri's is given here five plus 10 plus 20 and so on, and we are asking to find the sun off in terms off the geometric. Siri's now in this case and is eight. So we need to find the son of the 1st 8 terms. So for the Given Siri's, you may observe here that the first time which will call a one is equal to fire and the common racial are is equal to two because five times to 10 intentions to his 20 and soul winning and the number of terms he's. Eight. We're a team. So to find the some off any terms of the genetic Siri's used this far right here on s been is equal to anyone. Brackets one minus arteries, toe end bracket over one minus are where it's in. It presents the summer off in terms. A win presents the first term. I represent the colon ratio, and the end represents the number of terms he So to find the sun or age times here, the sum of age terms we'll need to substitute these values into the formal. So a one is fine. Bracket one minus two raised was our is to raised to the power of age bracket one. So over one minus R, which is to now, uh, use your calculator for computation here would find this is five rocket one minus two Race. The power eight is 256 deified, but remitted team for one. But in us, if you again If you use your con trait of computation you find that this son on the 1st 8 terms, uh, this series is equal to 1270 fine.

So here we want to find the sum of this geometric series and we know the ace up. One is equal to 20 ours equal to negative one over 10 and N is equal to 12. And we're gonna use the equation s up in is equal to a sub one times one minus R to the n over one minus r and that gives us when we begin. Our values s a 12 is equal to 20 times one minus negative. One over 10 to the 12 over. Oh, over one minus negative, one over 10. Is it a simplify that gives us 20 times one minus negative, one over 10? It's to the 12 over 11 over 10 and it's simple by that. Further, that gives us our answer of 18.18

Okay, so here we hear over magnetic cities. We need to find this somewhat the first pain times. So you're for us dummies minus three. Woman difference T That is the 2nd 0 minus four Stone. Your negative four plus three that is equal to minus one. Right? No, to some off enigmatic sequence that issuing by and by two multiply by the way, one plus and minus one times. You're an easy quarter, Dan. You're everyone is equal to minus three again Here and easy Board of den. And here, a common difference. D that support minus one. Stand by to his five to a multi level minus three. That is minor six 10 minus one. That is nine your nine month old baby minus one that is minus name, minus six minus nine. That is it for to get your 15 negative 15 murdered by fire. That disease Bordeaux Negative 75. Please disarm off the first tandems. What this given added medics equals Thank you

So We have problem by 21. And this we need to use formula to find the 1st 40 to find a summer 1st 40 terms of the series defined by an equal to five. And so we have been given and that is the number of terms as 40. And we have some formula for some of in terms and faster plus last term. But two. Okay. So he went Can be found out by just plugging one here. So 5.1 which is five A. N. That is a 40 will be five and 2 40 equal to 200. So let's plug in here in the formula As 40 will be equal to 40 and two five place 200 x two. So this is 20. So 20 and two 205 0014. So 4100 will be to answer Thank you.