Would you like to inspect the original subtitles? These are the user uploaded subtitles that are being translated: 1 00:00:00,480 --> 00:00:04,160 Today are going to be talking about how to calculate the first several terms of a sequence. 2 00:00:04,260 --> 00:00:08,940 And in this particular problem we've been given a 7th term of the sequence which is a seben is equal 3 00:00:08,940 --> 00:00:12,920 to 1 plus the quantity negative one half to the end. 4 00:00:12,990 --> 00:00:18,490 Now when we're asked to calculate the first terms of the sequence and the A7 term is this simple. 5 00:00:18,540 --> 00:00:21,960 All we need to do is plug in values of n. 6 00:00:21,990 --> 00:00:27,380 Starting with N equals 1 and simplifying to find the value of each term in the sequence. 7 00:00:27,380 --> 00:00:31,430 So for example we'll say here and is equal to 1. 8 00:00:31,440 --> 00:00:37,320 We're going to get the a sub one term of the sequence and that's going to be plugging in one for and. 9 00:00:37,340 --> 00:00:38,230 Will get one. 10 00:00:38,250 --> 00:00:45,540 Plus the quantity negative one half to the first power and then we're going to get a sub one is equal 11 00:00:45,540 --> 00:00:51,120 to positive one half or point 5 0 0 0. 12 00:00:51,120 --> 00:00:55,470 Often when we're capturing the first terms of the sequence we're asked to calculate to four decimal 13 00:00:55,470 --> 00:00:56,920 places or something like that. 14 00:00:57,000 --> 00:00:59,460 So we can go ahead and express it that way as well. 15 00:00:59,460 --> 00:01:04,380 And then we just follow the same process for the next terms in the sequence we say any equals two is 16 00:01:04,380 --> 00:01:10,660 going to give us a sub 2 which will be 1 plus negative 1 half squared. 17 00:01:10,680 --> 00:01:18,660 That's going to give us a sub 2 is equal to one plus one fourth or five fourths which is the same as 18 00:01:18,690 --> 00:01:22,270 1 point 2 5 0 0. 19 00:01:22,290 --> 00:01:25,950 We can continue on like this where we get an equals three. 20 00:01:26,100 --> 00:01:28,050 That's the ace of three term. 21 00:01:28,050 --> 00:01:36,930 That's one plus quantity negative one half to the third which gives us a sub 3 is equal to 1 minus 1 22 00:01:36,990 --> 00:01:44,280 eighth or just seven eighths which is the same as point 8 7 5 0. 23 00:01:44,430 --> 00:01:49,590 And I will keep going with all of the tedious writing but what you'll find is that if you keep going 24 00:01:49,620 --> 00:01:58,910 let's say to the first six terms of the sequence you'll get a sub 4 equals 17 over 16 or one point zero 25 00:01:58,950 --> 00:02:01,060 six to five. 26 00:02:01,260 --> 00:02:11,830 You'll get a sub 5 is equal to 31 over 32 which is also equal to point 9 6 8 8. 27 00:02:11,970 --> 00:02:23,370 And then you'll get a sub 6 is equal to 65 over 64 which is also one point 0 1 5 6 and you can just 28 00:02:23,370 --> 00:02:29,160 keep going like that if you want to try to use these terms to draw a conclusion about the limit of the 29 00:02:29,160 --> 00:02:31,560 sequence as and approaches infinity. 30 00:02:31,740 --> 00:02:37,110 You can go ahead and try to plot these values on a coordinate system and see if you can make a guess 31 00:02:37,170 --> 00:02:39,120 about the value they're approaching. 32 00:02:39,120 --> 00:02:41,430 I've gone ahead and done that already anyway. 33 00:02:41,520 --> 00:02:46,680 And what you can see is that if we plot these values and this is values for and here we found an equals 34 00:02:46,680 --> 00:02:49,590 1 2 3 and then 4 5 6. 35 00:02:49,590 --> 00:02:54,370 So this is the ace of one value here we found that with point five. 36 00:02:54,390 --> 00:02:56,190 So we plotted that point here. 37 00:02:56,220 --> 00:02:59,790 At one point five and we just kept going. 38 00:02:59,880 --> 00:03:06,270 And what you see is that these values oscillate back and forth across the line y equals one but they 39 00:03:06,270 --> 00:03:08,600 get closer and closer to that line. 40 00:03:08,610 --> 00:03:17,310 So the conclusion that we can then try to draw is that the limit as and goes to infinity of the sequence 41 00:03:17,700 --> 00:03:24,720 A saw then this entire sequence here is equal to 1 because as and goes farther and farther out to the 42 00:03:24,720 --> 00:03:29,700 right these points will approach the lying White was one more get closer and closer and closer to it 43 00:03:30,000 --> 00:03:31,080 without ever reaching it. 44 00:03:31,080 --> 00:03:35,910 So the limit is and approaches infinity of the entire sequence is one. 4879