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These are the user uploaded subtitles that are being translated: 1 00:00:00,380 --> 00:00:02,850 In this video we're talking about how to find the sum of a series. 2 00:00:02,850 --> 00:00:05,870 When the series is given to us in summation notation. 3 00:00:06,060 --> 00:00:08,910 So this each symbol right here it's called Sigma. 4 00:00:08,910 --> 00:00:15,110 This is what indicates a mation notation and the value here to the right of it is our series. 5 00:00:15,120 --> 00:00:21,800 So this value right here to the right of this sigma notation that is the series. 6 00:00:21,840 --> 00:00:28,740 So we often call this series a suburban so we could write this as a sub N is equal to quantity and minus 7 00:00:28,740 --> 00:00:30,650 one times quantity and plus two. 8 00:00:30,840 --> 00:00:32,500 That's our series. 9 00:00:32,580 --> 00:00:36,000 And what we've been asked to do is find the sum of this series. 10 00:00:36,000 --> 00:00:41,780 When we start at any quolls once where we start with the value given to us here this is called the index. 11 00:00:41,790 --> 00:00:47,550 So when we start and end equals 1 and we count up to the value that's given to us up here which in this 12 00:00:47,550 --> 00:00:48,460 case is 8. 13 00:00:48,630 --> 00:00:53,310 Oftentimes it will be infinity and we'll talk about infinite series later but here we're going to start 14 00:00:53,360 --> 00:00:59,190 an equals one and we're going to go up to eight and equals one means the first term or when n is equal 15 00:00:59,190 --> 00:01:04,500 to 1 and equals 2 would be the second term and equals three would be the third term et cetera until 16 00:01:04,500 --> 00:01:07,250 we get up to a value of N equals 8. 17 00:01:07,260 --> 00:01:09,420 The value here at the top. 18 00:01:09,420 --> 00:01:13,110 So if we want to find the sum of that series Here's what we do. 19 00:01:13,110 --> 00:01:17,460 We start by putting an end equals 1 and then we know we're going to plug into 3 4 all the way up to 20 00:01:17,460 --> 00:01:17,760 8. 21 00:01:17,760 --> 00:01:20,370 Let's just go ahead and write those out here. 22 00:01:20,700 --> 00:01:25,110 Four five six seven and eight. 23 00:01:25,290 --> 00:01:26,500 OK so we're going to start with. 24 00:01:26,520 --> 00:01:27,620 Any equals 1. 25 00:01:27,630 --> 00:01:30,980 And we're going to plug any one into our series here a 7. 26 00:01:31,080 --> 00:01:34,220 So we're going to get is 1 minus 1 or 0. 27 00:01:34,230 --> 00:01:38,580 So we're going to get ZERO times one plus two which we know is three. 28 00:01:38,580 --> 00:01:45,300 So when we plug in N equals 1 we're going to get 0 times 3 then we're going to add to that whatever 29 00:01:45,300 --> 00:01:47,370 we get when we plug in N equals 2. 30 00:01:47,520 --> 00:01:52,410 Well here we're going to get to minus 1 or 1 and here we're going to get two plus two which is for us 31 00:01:52,410 --> 00:01:55,350 we get 1 times 4 then we're just going to keep going. 32 00:01:55,350 --> 00:01:59,630 Now we're going to plug in N equals three and we're going to do this all the way up to N equals 8. 33 00:01:59,640 --> 00:02:04,740 So plugging in three we get three minus one is two three plus two is five. 34 00:02:04,830 --> 00:02:10,620 Plugging in four we're going to get four minus one is three four plus two is six plugging in five we're 35 00:02:10,620 --> 00:02:18,990 going to get five minus one is four five plus two is seven plugging in six six minus one is five six 36 00:02:18,990 --> 00:02:22,410 plus two is eight plugging in seven. 37 00:02:22,440 --> 00:02:27,060 We're going to get seven minus one is six seven plus two is nine. 38 00:02:27,180 --> 00:02:28,430 And then plugging in eight. 39 00:02:28,470 --> 00:02:32,840 We're going to get eight minus one is seven eight plus two is ten. 40 00:02:32,910 --> 00:02:37,650 And notice here that when you expand the terms of this series from end equals one to end equals eight 41 00:02:37,950 --> 00:02:39,930 you start to see a pattern. 42 00:02:39,930 --> 00:02:47,020 So for this and minus one term you can see we have 0 1 2 3 4 5 6 and 7. 43 00:02:47,070 --> 00:02:48,820 And for this end plus 2. 44 00:02:48,840 --> 00:02:54,060 We end up with 3 4 5 6 7 8 9 10. 45 00:02:54,060 --> 00:02:58,890 So once you notice the pattern you don't have to calculate each value for each term. 46 00:02:58,890 --> 00:03:02,480 You can just follow the pattern until you get up to whatever the last term is. 47 00:03:02,480 --> 00:03:04,270 In this case and equals eight. 48 00:03:04,380 --> 00:03:07,400 So now we just want to simplify 0 times 3 is zero. 49 00:03:07,430 --> 00:03:13,190 So that's going to become 0 and go away 1 times 4 is for two times five is ten. 50 00:03:13,200 --> 00:03:20,010 Three times six is eighteen four times seven is twenty eight five times eight is forty six times nine 51 00:03:20,010 --> 00:03:21,060 is fifty four. 52 00:03:21,060 --> 00:03:26,830 And here we get 70 when we add all those together we end up with two hundred twenty four. 53 00:03:26,910 --> 00:03:32,370 So we can say that the sum of this series when the series is quantity and minus one times quantity and 54 00:03:32,370 --> 00:03:38,130 plus two the sum of the series from end equals 1 to and equals eight or the sum of the series through 55 00:03:38,130 --> 00:03:41,340 its first eight terms is 224. 5815