00:00Alright everybody, so in this
00:01lesson we're going to go
00:02through arithmetic sequences and series.
00:05Now firstly, what is an
00:07arithmetic sequence? Well, if we,
00:09let's say we do 10,
00:1313, 16, 19, etc. This
00:17is an arithmetic sequence because
00:19it's going up in threes.
00:21So 10 plus 3 is
00:2313 plus 3 is 16
00:24plus 3 is 19. An
00:25arithmetic series is just when
00:27we add them.
00:2810 plus 13 plus 16
00:31plus 19. Now we're given
00:33these two formulae in the
00:34formulae booklet. This is the
00:35nth term of an arithmetic
00:37sequence and this is the
00:38sum of n terms of
00:39an arithmetic sequence. So this
00:40is basically the sum of
00:41an arithmetic series. So over
00:46here I've written what these
00:47letters stand for. So u
00:48of n is the nth
00:50term. U1 is the first
00:51term and is the number
00:52of terms. D is the
00:54common difference. So this
00:56When I add 3 to
00:59get to here, the common
01:01difference is 3 plus 3
01:04plus 3. Now you can
01:07also have an arithmetic sequence
01:10that goes down like 20,
01:1415, 10, etc. And the
01:19common difference here is negative
01:215. So if it's going
01:24down
01:24in five or whatever then
01:26the common difference is just
01:27negative whatever it's going down
01:28by and S of n
01:30is then the sum of
01:31n terms. Okay I'm going
01:34to go straight into the
01:34example and go through how
01:38we use these formulae to
01:39solve a given problem. So
01:42Fred is training for a
01:43marathon. He decides to run
01:455 ,000 meters on his
01:46first day of training. He
01:48then increases the distance he
01:49runs by 500 by 300
01:51meters each day.
01:52How far does Fred run
01:54on the fifth day? Okay,
01:56the first thing I like
01:57to do in these situations
01:59is when when we're dealing
02:00with sequences right out the
02:02sequence What is the sequence?
02:03Well, it's 5000 in the
02:05first day then it's gonna
02:06be 5 ,300 then it's
02:09gonna be 5 ,600 then
02:125 ,900 and actually if
02:15we keep going We'll get
02:17to 6 ,200 which is
02:18what he runs in the
02:20fifth day
02:20one two three four five
02:23keep going six thousand five
02:26hundred now for part eight
02:28there's no problem if writing
02:31out the first five terms
02:33it write them all out
02:34and then say this is
02:35what he runs in the
02:35fifth term that is the
02:37fifth day that is correct
02:38however what if it said
02:40how far does he run
02:41on the I don't know
02:43seven hundred and sixty second
02:46day that's not a zem
02:48That's not as easy to
02:49do because you don't want
02:50to write out 700 numbers
02:52or whatever it happens to
02:53be. So I'm going to
02:53show you a kind of
02:54more technical way to do
02:56this. So A, what we're
02:58trying to find is how
03:00far you run on the
03:00fifth date, which is the
03:02fifth term. So this is
03:05U of five, we're trying
03:07to find. So let me
03:07write down the formula first.
03:09U1 equals, sorry, Un equals
03:11U1 plus n minus 1d.
03:16Okay.
03:16So we're trying to find
03:18the nth term, not the
03:20sum, because we're not adding
03:21up how much he runs
03:22on each day. I like
03:25to at the side, write
03:26out what each of these
03:27things are. So u of
03:28n, let me write down
03:34u of 5 is what
03:37we don't know. u of
03:381 is 5 ,000.
03:44is then 5 because we
03:47want to find what he's
03:47running out of the fifth
03:48day and d we know
03:50is 300 because it's increasing
03:52by 300 each day. So
03:54u of 5 equals u
03:561, 5 ,000 plus n
04:00minus 1d which is 5
04:03minus 1 times 300 so
04:07n minus 1d minus n
04:09minus 1 times d. This
04:12is 5
04:12I think this is my
04:14calculator, but I don't even
04:15need to. 5 ,000 plus
04:16four times 300 is 1
04:18,200, giving me 6 ,200,
04:22which is what I expected.
04:25So that's how we apply
04:25the kind of, well, the
04:27end term formula. Great. On
04:31which day will Fred run?
04:33Will Fred first run more
04:35than 21k or 21 kilometers?
04:39Let's have a think about
04:40that.
04:40B. Which day will he
04:45first run more than 21
04:47kilometers? Okay, we're still dealing
04:48with the, we're still dealing
04:50with the nth term. So
04:53it's u of n equals
04:57u1 plus n minus 1
05:01d. But we now know
05:06u of n, it's u
05:08of n is
05:0821 ,000. So what we're
05:10going to do is we're
05:11going to solve the equation
05:12and find when does n
05:14equal 21? Or when does
05:16u of n equal 21
05:17,000? Because it tells me
05:19that's 21 kilometers, which is
05:2121 ,000 meters. So when
05:24the thing I don't know
05:26is n. So let me
05:27write this down. U of
05:29n is 21 ,000. U
05:33of 1 is still 5
05:35,000.
05:36n is what I don't
05:38know this is what we're
05:39trying to find out and
05:41d I do know it's
05:43still 300 so I'm going
05:45to do u1 5000 plus
05:49n minus 1 times d
05:53which is times 300 okay
05:56let's do this on the
06:00calculator I could I could
06:03rearrange it to subtract 5000
06:04divided by
06:04300 add one, but let's
06:08use numerical solve because it's
06:11so nice. So we have
06:1321 ,000 equals 5 ,000
06:18plus n minus one times
06:24300. And then we have
06:27to do comma n. So
06:30n solve type out your
06:31equation and then you press
06:32comma
06:33And as you tell your
06:34calculator, find and please present
06:37it and we get 54
06:38.3333 and equals 54 .3333.
06:45So he runs exactly 21
06:47,000 on the 54 .3333
06:50day, which doesn't really make
06:51sense. But what that means
06:52is on the 54th day,
06:55he runs less than 21k
06:56and on the 55th day,
06:58he'll run
07:01more than 21k. So I'm
07:03going to say therefore first
07:05day greater than 21 kilometers
07:09is the 55th, 55th day.
07:19Okay, next question. I'll do
07:23it down here. Well, here.
07:26If Fred continues this training,
07:28a schedule
07:29For 30 days calculate the
07:30total distance in kilometers he
07:33will run. Okay, let's actually
07:34do this question over here
07:36on the side. Okay, part
07:42C. So he's going to
07:44continue running for 30 days,
07:47calculate the total distance in
07:49kilometers he will run. So
07:50what we're looking at is
07:52the sum, the S of
07:55N formula, S of N.
07:57and equals. Now, there's two
07:59formula for S of n
08:01or the sum of n,
08:02sum of n terms, this
08:03one and this one. This
08:05one we use when we
08:06know the nth term. When
08:09you know u of n,
08:10just use this one. It's
08:11the first one plus the
08:12last one times n divided
08:13by two. This one, use
08:18if we don't know u
08:20of n. In fact, they're
08:21basically the same formula except
08:23this has been subding
08:25into this to give you
08:26this one. So let me
08:28summarize, use this one if
08:30you know u of n
08:31and use this one if
08:32you don't know u of
08:32n. I don't know u
08:34of n because I don't
08:36know what u of 30
08:38is. I don't know how
08:38much you runs on this
08:39on the 30th day. So
08:41I'm going to use this
08:42one. This is the more
08:43common one you end up
08:44using. So it's n over
08:452 times 2 u 1
08:49plus n minus 1 d.
08:52OK, I'm going to write
08:52out the thing
08:53that I know or that
08:57I don't know. So S
08:58of N, I don't know.
09:00That's what I'm trying to
09:01find. Or S of 30,
09:02I should say, really. S
09:04of 30, I don't know.
09:08N equals 30 because he's
09:12going to run for 30
09:13days. U1 is the first
09:17term that is going to
09:18be 5 ,000, still 5
09:20,000.
09:21And D, let's put a
09:24bracket here, and D is
09:26going to be 300. Let's
09:29still stay the same. Okay,
09:32so now I just subbing
09:33these values. S30 equals 30
09:37over two brackets. Two, U1
09:41is 5 ,000 plus N
09:46minus one, so N is
09:4830,
09:4930 minus 1 times, let
09:53me do that again, times
09:56300. Okay, just fits equals
10:01now. I'm going to put
10:02all this into my calculator
10:04so I can see it.
10:07Okay, I'm going to do
10:1030 over 2, put it
10:12in exactly as I see
10:12it. I could have just
10:13done 15, but why not?
10:16Two times 5 ,000.
10:172 times 5 ,000 plus
10:2630 minus 1. Now, again,
10:32you could obviously put 29
10:35there, but I'm going to
10:35do 30 minus 1 times
10:39300. OK, I think I've
10:42done two brackets there, but
10:44that won't make a difference.
10:45Okay, press enter and I
10:48get 280 ,500, 280 ,500.
10:53So that's what he's run.
10:55Let's say for a month,
10:57he's run it in the
10:58whole month. He's run 280
11:00,000. Well, hang on a
11:03second. He's run 280 ,000
11:06meters. Be very careful. It
11:08says, calculate the total distance
11:10in kilometers he will run.
11:13So if he's run
11:13around 280 ,000 meters, this
11:16equals 280 .5 kilometers. So
11:22that's when we add up
11:24all the number of kilometers
11:26he runs in. The first
11:27day, this plus this plus
11:28this plus this plus this
11:29plus this all the way
11:29up for 30 days. And
11:31you may think, well, that
11:32seems like a crazy amount
11:33to run. But if you're
11:35training for a marathon, yeah,
11:37you probably should be running
11:38about 70 kilometers a week,
11:39at least.
11:41Yeah, that makes sense. Okay,
11:46so that's arithmetic sequences in
11:47series. Of course, you need
11:49to understand what an arithmetic
11:51sequence is and then that
11:53the series is just the
11:54sum of N terms. This
11:56is your typical kind of
11:57question that you can be
11:58asked, but obviously the IB
12:00will, well, they can ask
12:02questions in whatever way they
12:03want. So make sure you
12:04go out and practice the
12:05past papers. Okay, see you
12:07in the next lesson when
12:08we are doing
12:10geometric sequences in series.