Arithmetic Sequences and Arithmetic Series - Basic Introduction - Free Educational videos for Students in K-12 | Lumos Learning

Arithmetic Sequences and Arithmetic Series - Basic Introduction - Free Educational videos for Students in k-12


Arithmetic Sequences and Arithmetic Series - Basic Introduction - By The Organic Chemistry Tutor



Transcript
00:00 in this video , we're going to focus mostly on
00:02 everything . Tick sequences now to understand what an arithmetic
00:07 sequences . It's helpful to distinguish it from a geometric
00:10 sequence . So here's an example of number from six
00:15 sequence . The numbers 37 11 15 , 19 23
00:24 and 27 represents an arithmetic sequence . This would be
00:31 a geometric sequence . three six , 12 24 48
00:39 96 1 92 . Do you see the difference between
00:45 these two sequences ? And do you see any patterns
00:48 within them and the arithmetic sequence on the left ?
00:53 And notice that we have a common difference . This
00:55 is the first term . This is the second term
00:58 . This is the 3rd , 4th and 5th term
01:01 to go from the first term to the second term
01:04 . We need to add for to go from the
01:07 second to the third term we need to add for
01:11 . And that is known as the common difference in
01:19 the geometric sequence . You don't have a common difference
01:21 . Rather you have something that is called the common
01:24 ratio . To go from the first term to the
01:27 second term . You need to multiply by two to
01:31 go from the second term to the third term .
01:33 You need to multiply by two again . So that
01:36 is the R . Value . That is the common
01:38 ratio . So in an arithmetic sequence the pattern is
01:44 based on addition and subtraction . In a geometric sequence
01:48 , the pattern is based on multiplication and division .
01:53 Now the next thing that when you talk about is
01:56 the mean , how to calculate the arithmetic mean and
01:59 the geometric mean . The arithmetic mean is basically the
02:03 average of two numbers . It's A Plus B divided
02:07 by two . So when taken an arithmetic mean of
02:12 two numbers within an arithmetic sequence , let's say .
02:15 If we were to take The mean of three and
02:18 11 we would get the middle number in that sequence
02:21 . In this case we would get seven . So
02:24 if you would add three plus 11 and divide by
02:28 23 plus 11 is 14 . 14 divided by two
02:31 gives you seven . Now let's say if we wanted
02:34 to find The emergency meeting between seven and 23 it's
02:40 going to give us the middle number Of that sequence
02:43 which is 15 . So if you would add up
02:46 seven Plus 23 divided by two , seven plus 23
02:51 is 30 30 divided by two is 15 . So
02:55 that's how you can calculate the arithmetic mean and that's
02:57 how you can identify it within In November six sequence
03:04 . The geometric mean is the square root Of eight
03:08 times being . So let's say if we want to
03:11 find the geometric mean between three and six , It's
03:15 going to give us the middle number of the sequence
03:17 . Which is I mean if we were to find
03:20 the geometric mean between three and 12 We will get
03:23 the middle number of that sequence which is six .
03:28 So in this case a history be is 12 .
03:31 Three times 12 is 36 . The square root of
03:34 36 is six . Now let's try another example .
03:39 Let's find the geometric mean Between six and 96 .
03:44 This should give us the Middle # 24 . Now
03:52 we need to simplify this . Radical 96 is six
03:59 times 16 . six times 6 is 36 . The
04:05 square root of 36 is six . The square to
04:07 16 is four . So we have six times 4
04:12 Which is 24 . So as you can see the
04:15 geometric mean of two numbers within the geometric sequence will
04:19 give us the middle number in between those two numbers
04:21 in that sequence . Now , let's clear away a
04:25 few things . The formula that we need to find
04:32 , the f term of an arithmetic sequence is a
04:36 seven is equal to a someone Plus and -1 times
04:41 the common difference d . In a geometric sequence it's
04:45 a suburban is equal to a one times are Race
04:50 to the N -1 . Now let's use that equation
04:59 to get the fifth term in arithmetic sequence . So
05:04 that's going to be a 75 . A sub one
05:08 is the first term which is three , N is
05:13 five . Since we're looking for the fifth term ,
05:14 the common difference is four . In this problem 5
05:20 -1 is four . four times 4 . 16 .
05:24 3 plus 16 is 19 . So this formula gives
05:31 you any terms in the sequence . You could find
05:33 the fifth term , the seventh term , the 100
05:36 term and so forth . Now in a geometric sequence
05:46 , we could use this formula . So let's calculate
05:49 the 6th term Of the geometric sequence . It's going
05:52 to be a sub six Which equals a sub one
05:55 . The first from history . The common ratio is
05:58 too And this is gonna be raised to the 6
06:02 -1 . 6 -1 is five and then two to
06:07 the fifth power . If you multiply 25 times two
06:10 times two times two times two times two . So
06:15 we can write it out . So this here that's
06:17 for three twos . Make 84 times eight is 32
06:22 . So this is three times 32 . Three times
06:26 30 is 93 times two is 6 . So this
06:30 will give you 96 . So that's how you can
06:35 find the F term in a geometric sequence . By
06:41 the way , make sure you have a sheet of
06:43 paper to write down these formulas . So that when
06:46 we work on some practice problems , you know what
06:49 to do now the next thing we need to do
07:05 is be able to calculate the partial sum of a
07:08 sequence . S seven is the partial sum of a
07:15 series of a few terms and it's equal to the
07:19 first term plus the last term Divided by two times
07:24 . And For geometric sequence the partial sum s event
07:28 is going to be a sub one Times 1 -
07:32 are basically and Over 1 -2 . So let's find
07:38 the sum of the first seven terms in the sequence
07:43 . So that's going to be S sub seven .
07:46 That's going to equal the first turn plus the 7th
07:48 turn divided by two times End . Where N .
07:54 is the number of terms which is seven . Now
07:57 think about what this means . So basically to find
08:00 the sum of an arithmetic sequence , you're basically taking
08:04 the average of the first and the last term in
08:08 that sequence and then multiplying it by the number of
08:11 terms in that sequence . Because this is basically the
08:15 average of three and 27 . And we know the
08:21 average or the arithmetic mean of three and 27 .
08:24 That's gonna be the middle number 15 . So let's
08:27 go ahead and plug this in . So this is
08:29 3-plus 27 . Over to times seven . Three plus
08:35 27 is 30 plus two . I mean well 30
08:38 divided by two . That's 15 . So the average
08:42 Of the first and last term is 15 times seven
08:46 . 10 times seven is 75 times seven is 35
08:50 . So this is gonna be one of five .
08:53 So that's the some of the first seven terms .
09:00 And you can confirm this with your calculator if you
09:02 add up three plus seven plus 11 plus 15 Plus
09:08 19 plus 23 And then plus 27 . And that
09:13 will give you s . f . seven . The
09:15 some of the first seven terms . Go ahead and
09:22 add up those numbers . If you do you'll get
09:25 one of five . So that's how you can confirm
09:28 your answer . Now let's do the same thing with
09:33 a geometric sequence . So let's get this some Of
09:38 the 1st 6 terms As sub six . So this
09:43 is going to be three Plus six plus 12 plus
09:48 24 plus 48 plus 96 . So we're adding the
09:55 first six terms now because it's not many terms were
10:04 added we can just simply plug this into our calculator
10:08 And we'll get 189 . But now let's confirm this
10:11 answer using the formula . So as sub 6 to
10:16 some of the first six terms Is equal to the
10:19 first term . A sub one which history Times one
10:24 -R . R . is the common ratio . Which
10:25 is to race to the end and it's six over
10:30 one minus R . Or one minus two . I'm
10:33 gonna work over here since this more space Now 2
10:37 6 , That's gonna be 64 . If you recall
10:41 two to the fifth , power is 32 . If
10:43 you multiply 32 x two you get 64 . So
10:48 this is gonna be 1 -64 And 1 -2 is
10:52 -1 . So this is three times 1 -64 Is
11:01 -63 . So we could cancel the two negative science
11:05 A negative divided by a negative will be a positive
11:08 . So this is just three times 63 , three
11:10 times six is 18 . So three times 60 has
11:14 to be 1 80 and then three times three is
11:17 91 80 plus nine Adds up to 189 . So
11:23 we get the same answer . Now , what is
11:27 the difference between a sequence in a series ? I'm
11:32 sure you heard of these two terms before . But
11:34 what is the difference between them ? Now ? We've
11:37 already considered what ? And the different sects sequences .
11:41 A sequence is basically a list of numbers . So
11:47 that's a sequence . A series is the sum of
11:51 the numbers in the sequence . So this here is
12:01 and arithmetic sequence . This is an arithmetic series because
12:06 it's the sum of an arithmetic sequence . Now ,
12:28 what we have here is a sequence but it's a
12:31 geometric sequence as we've considered earlier . This is a
12:37 geometric series . It's the sum of a geometric sequence
12:42 . Now there are two types of sequences and two
12:45 types of series . You have a finite sequence and
12:49 an infinite sequence and is also a finite series .
12:52 In an infinite series . This sequence is finite ,
12:59 it has a beginning and it has an end .
13:01 This series is also finite . It has a beginning
13:04 and has an end . In contrast , if I
13:07 would write 37 11 , 15 19 and then dot
13:12 dot dot this would be an infinite sequence . The
13:20 presence of these dots tells us that the numbers keep
13:23 on going to infinity . Now the same is true
13:28 for serious . Let's see if I had three plus
13:38 seven Plus 11 plus 15 plus 19 and then plus
13:44 dot dot dot dot dot . That would also be
13:48 an infinite serious . So now , you know the
13:51 difference between the finite series and an infinite series .
13:55 Now , let's work on some practice problems , described
13:58 the pattern of numbers shown below . Is it a
14:02 sequence or serious ? Is it finite or infinite ?
14:06 Is it arithmetic , geometric or neither ? So let's
14:11 focus on if it's a sequence or series . 1st
14:15 part eight . So we got the numbers 4 ,
14:18 7 , 10 , 13 , 16 , 19 .
14:21 We're not adding the numbers were simply making a list
14:24 of it . So this is a sequence . The
14:30 same is true . For part B will simply listing
14:33 the numbers . So that's a sequence in part C
14:36 we're adding a list of numbers . So since we
14:39 have a some this is going to be a series
14:42 D . Is also a series E . That's a
14:47 sequence for F . We're adding numbers . So that's
14:51 a series and the same is true Fiji . So
14:56 hopefully this example helps you to see the difference between
14:58 a sequence in the series . Now let's move on
15:01 to the next topic . Is it finite or is
15:05 it infinite to answer that ? All we need to
15:11 do is identify if we have a list of dots
15:14 at the end or not Here . This ends at
15:17 19 . So that's a finite sequence . The dots
15:23 here tells us it's going to go forever . So
15:24 this is an infinite sequence . This one we have
15:32 the dot . So this is going to be an
15:33 infinite series . This ends at 162 , so it's
15:40 finite . So we have a finite series . This
15:44 is gonna be an infinite sequence . Next we have
15:51 an infinite series And the last one is a finite
15:56 serious . Now let's determine if we're dealing with and
16:03 a different take geometric or neither sequences series . So
16:09 we're looking for a common difference or common ratio .
16:13 So for a notice that we have a common difference
16:16 of 34 plus three is 77 plus three is 10
16:21 . So because we have a common difference , this
16:24 is going to be and arithmetic sequence for be going
16:32 from the first number two . The second number we
16:34 need to multiply by 24 times two is 88 times
16:38 two is 16 . So we have a common ratio
16:42 which makes this sequence geometric yeah for answer choice C
16:51 . Going from 5 to 9 that's plus four And
16:54 from 9 to 13 that's plus four . So we
16:57 have a common difference . So this is going to
17:00 be not a arithmetic sequence but and arithmetic series for
17:07 answer choice D . Going from 2 to 6 were
17:10 multiplied by three And then six times street is 18
17:15 . So that's a geometric the geometric series . Now
17:22 for e Going from 50 to 46 that's a difference
17:26 of negative four And 46 - 42 . That's a
17:30 difference of -4 . So this is arithmetic for f
17:39 . We have a common ratio of 43 times four
17:42 is 12 . 12 times four is 48 . And
17:47 if you're wondering how to calculate D . N .
17:49 R . To calculate D take the second term subtracted
17:52 by the first term 7 -4 Street . Or you
17:56 can take the third term subtracted by the second .
17:59 10 -7 is 4th . In the case of f
18:02 if you take 12 divided by three , you get
18:04 four 48 divided by 12 to get four . So
18:08 that's how you can calculate the common difference or the
18:10 common ratio . It's by analyzing the second term with
18:14 respect to the first one . So since we have
18:17 a common ratio , this is gonna be geometric Fergie
18:25 . If we subtract 18 by 12 we get a
18:27 common difference of positive 6 20 for minus 18 Gives
18:32 us the same common difference of six . So this
18:36 is going to be arithmetic . So now let's put
18:40 it all together , let's summarize the answers . So
18:44 for part a what we have is a finite of
18:47 reference six sequence . Part B . This is an
18:51 infinite geometric sequence . See we have an infinite arithmetic
18:56 series . D is a finite geometric series . E
19:02 . Is an infinite referencing sequence . F is an
19:06 infinite geometric series . G is a finite arithmetic series
19:13 . So we have three columns of information with two
19:16 different possible choices . Thus to to the third is
19:20 eight , which means that we have eight different possible
19:22 combinations . Right now , I have seven out of
19:26 the eight different combinations . The last one is a
19:29 finite geometric sequence , which I don't have listed here
19:34 . So now , you know how to identify whether
19:37 you have a sequence or series If it's arithmetic or
19:40 geometric and if it's finite or infinite number , two
19:45 Rights of first four terms of the sequence defined by
19:48 the Formula A seven is equal to three and -7
19:54 . So the first thing we're going to do is
19:56 find the first term . So we're gonna replace end
19:59 with one . So it's gonna be 3 -7 which
20:04 is negative for . And then we're going to repeat
20:06 the process . We're going to find the second term
20:08 a sub two . So it's three times 2 -7
20:13 Which is -1 . Next we'll find a sub three
20:20 . three times 3 is 9 -7 . That's too
20:22 . And then the fourth term eights up four ,
20:26 That's going to be 12 -7 which is five .
20:30 So we have a first term of negative four then
20:33 it's negative one 25 and then the sequence can continue
20:40 . So the comment difference in this problem is positive
20:43 three . going from negative 1-2 . If you add
20:46 three you'll get to And then 2-plus 3 is five
20:52 . But this is the answer for the problem .
20:54 So this is those are the first four terms of
20:58 the sequence number three . Right ? The next three
21:02 terms of the following arithmetic sequence . In order to
21:08 find the next re terms , we need to determine
21:11 the common difference . A simple way to find the
21:15 common difference is to subtract the second term by the
21:18 first term , 20 to -15 is seven . Now
21:26 , just to confirm , we need to make sure
21:27 that the difference between the third and the second term
21:30 is the same . 29 -22 . There's also seven
21:37 . So we have a common difference of seven .
21:39 So we could use that to find the next three
21:41 terms . So 36 plus seven is 43 43-plus 7
21:47 is 50 50 plus seven is 57 . So these
21:51 are the next three terms of the arithmetic sequence .
21:55 Here's a similar problem but presented differently . Right ?
21:59 The first five terms of an arithmetic sequence Given a
22:03 one in D . So we know the first term
22:07 is 29 and the common difference is negative for .
22:11 So this is all we need to write the first
22:14 five terms . If the common difference is negative for
22:17 then the next term is gonna be 29 plus negative
22:19 four , which is 25 25 Plus -4 or 25
22:25 -4 is 21 . 21 , -4 , 17 17
22:30 -4 is 13 . So that's all we need to
22:32 do in order to write the first five terms of
22:36 the romantic sequence . Given this information number five ,
22:41 right ? The first five terms of the sequence defined
22:44 by the following recursive formulas . So let's start with
22:49 the first one part A . So we're given the
22:53 first term . What are the other terms when dealing
22:58 with recursive formulas we need to realize is that you
23:01 get the next term by plugging in the previous term
23:05 . So let's say . And this too When N
23:08 is too this is a sub two and that's going
23:12 to equal a sub minus one to minus one is
23:15 one . So this becomes a sub one plus four
23:20 . So the second term is going to be the
23:22 first term three plus four , which is seven .
23:29 So we have three as the first term , seven
23:32 as a second terms and that let's find the next
23:34 one . So let's plug in three for end .
23:37 So this becomes a century , The next one .
23:43 This becomes a sub three modest one or a sub
23:46 two plus four . So this is seven Plus four
23:53 , which is 11 . At this point we can
23:56 see that we have an arithmetic sequence with a common
23:58 difference of four . So they get the next two
24:01 terms we could just add for It's gonna be 15
24:04 and 19 . So that's it for part eight .
24:11 So when dealing with recursive formulas , just remember you
24:14 get your next term by using the previous term now
24:18 for part B it's gonna be a little bit more
24:20 work . So plugging in and equals two . We
24:25 have the second term It's going to be three times
24:29 the first term plus two . The first term is
24:33 to so three times two is six plus two .
24:36 That gives us eight . So now let's plug in
24:40 an equal stream . When industry we have this equation
24:47 , a submarine is equal to three times a sub
24:49 2-plus 2 . So we're going to take eight and
24:54 plug it in here to get the third term .
24:57 So it's three times eight plus two . three times
25:01 8 is 24 plus two . That's 26 . Now
25:08 let's focus on the fourth term when N is four
25:10 . So this is going to be a sub four
25:12 is equal to three times a sub three plus two
25:17 . So now we're gonna plug in 26 for a
25:19 sub three . So it's three times 26 plus two
25:27 . Three times 26 is 78 plus two , that's
25:30 going to be 80 . Now let's focus on the
25:36 5th term . So a sub five is going to
25:40 be three times a sub four plus two . So
25:45 that's three times 80 plus two . Three times eight
25:49 is 24 . So three times 80 is to 40
25:52 plus two . That's going to be 242 to the
25:59 first five terms are too eight , 26 80 And
26:07 to 42 . So this is neither and arithmetic sequence
26:12 . Nor is it a geometric sequence , Number six
26:17 . Right . A general formula or explicit formula ,
26:21 which is the same for the sequence is shown below
26:25 in order to write a general formula or an explicit
26:27 formula . All we need is the first term and
26:30 the common difference if it's an arithmetic sequence which for
26:34 part A it definitely is . So if we subtract
26:39 14 by eight we get six and if we subtract
26:42 20 by 14 we get six . So we can
26:44 see that the common difference Is positive six and the
26:49 first term is eight . So the general formula is
26:54 a seven is equal to a someone Plus N -1
26:58 times deep . So all we need is the first
27:01 term and the common difference and we can write a
27:05 general formula or an explicit formula . The first term
27:10 is eight D . Is six . Now what we're
27:14 gonna do is we're going to distribute six to end
27:16 -1 . So we have six times then which is
27:22 six n . And then this will be -6 .
27:25 Next we need to combine like terms , so eight
27:28 plus negative six or eight minus six that's going to
27:31 be positive too . So the general formula is six
27:36 n plus two . So if we were to plug
27:42 in one this will give us the first term 86
27:46 times one plus two is eight . If we were
27:49 to plug in four it should give us the fourth
27:51 term 26 . 6 times four is 24 Plus two
27:56 , that's 26 . So now that we have the
28:00 explicit formula for part A what about the sequence in
28:03 part B . What should we do if we have
28:07 fractions ? If you have a fraction like this or
28:13 a sequence of fractions and you need to write an
28:16 explicit formula , try to separate it into two different
28:20 sequences . Notice that we have in arithmetic sequence .
28:24 If we focus on the numerator , that sequences two
28:29 , 3 , 4 , 5 and six . For
28:34 the denominator we have the sequence 357911 . So for
28:40 the sequence on top the first term is to and
28:43 we can see that the common differences one , The
28:45 numbers are increasing by one . So using the formula
28:49 A seven is equal to a sub one plus n
28:52 minus one times . D . We have that .
28:55 A sub one is 2 & D is one .
29:00 If you distribute one to end -1 you're just going
29:02 to get N -1 . So we can combine two
29:06 and negative one Which is positive one . So we
29:10 get the formula and plus one . Yeah . And
29:15 you can check it when you plug in 11 plus
29:17 one is two . So the first term is to
29:21 If you were to plug in five five plus one
29:24 at six that will give you the fifth term which
29:27 is six . Now let's focus on the sequence of
29:32 the denominators . The first term mystery . The common
29:37 difference we could see us too . 5 -3 is
29:40 2 7 -5 is too . So using this formal
29:44 again we have a seven is equal to a sub
29:48 one . A sub one history plus And -1 times
29:52 d . d . s . two . So let's
29:56 distribute to to end -1 . That's gonna be two
30:00 N -2 . And then let's combine like terms 3
30:05 -2 is a positive one . So a seven is
30:10 going to be two N plus one . So if
30:18 we want to calculate the first term we plug in
30:21 one for n two times one is two plus one
30:24 . It gives us three If we want to calculate
30:27 the 4th term and it's four . Two times four
30:30 is eight plus one . It gives us not .
30:35 So you always want to double check your work to
30:36 make sure that you have the right formula . So
30:39 now let's put it all together . So we're going
30:48 to write a seven and we're going to write as
30:51 a fraction . The sequence for the numerator is n
30:54 plus one . The sequence for the denominator is two
30:58 , n plus one . So this right here represents
31:08 the sequence that corresponds to what we see in part
31:11 B . And we can test it out . Let's
31:16 calculate the value of the third term . So let's
31:20 replace and with three it's going to be three plus
31:22 one Over two times ST plus one . three plus
31:27 1 is four , two times three is six plus
31:30 one at seven . So we have four of the
31:32 seven . If we wish to calculate the fifth term
31:35 , it's going to be five plus one over to
31:39 Times five Plus 1 . Five plus one is 62
31:42 times five is 10 plus one . That's 11 .
31:48 And so any time you have to write an explicit
31:50 formula given a sequence of fractions , separate the numerator
31:55 and there has not been into two different sequences .
31:58 Hopefully they're both arithmetic . If it's geometric , you
32:01 may have to look at another video that I'm going
32:03 to make soon on geometric sequences but break it up
32:07 into two separate sequences and then write the formulas that
32:09 way and then put the two formulas in a fraction
32:12 and that's how you can get the answer number seven
32:16 , write a formula for the end of term of
32:18 the arithmetic sequences shown below . Surviving . The formula
32:24 for the f term is basically the same as writing
32:26 a general formula for the sequence or an explicit formula
32:31 . So we need to identify the first term which
32:33 we could see us five . And the common difference
32:37 14 -5 is nine 203 -14 is nine as well
32:44 . So once we have these two we can right
32:46 the general formula . So let's replace the first term
32:53 a someone with five And let's replace d . with
32:57 nine . Now Let's distribute 9 to end -1 .
33:05 So we're going to have nine N -9 . Next
33:08 let's combine like terms . So it's going to be
33:13 nine n . and then 5 -9 is -4 .
33:19 So this is the formula for the f term of
33:22 the sequence . Now let's do the same for part
33:33 B . So the first term is 150 . The
33:38 common difference is going to be 143 -150 which is
33:42 -7 . To confirm that if you subtract 1 36
33:46 by 1 43 you also get negative seven . Now
33:53 let's plug it into this formula to write the general
33:55 equation . So a seven is going to be 150
34:01 plus And -1 times d . Which is -7 .
34:06 So let's distribute negative 7 to end -1 . So
34:12 it's gonna be 1 50 minus seven . End and
34:15 then negative seven times negative one . That's going to
34:18 be positive seven . So a sub N is going
34:21 to be negative seven N Plus 1 57 . Or
34:27 you could just write it as 1 57 minus seven
34:33 N . So that is the formula for the term
34:38 of the arithmetic sequence . Now let's move on the
34:45 part beat Calculate the value of the 10th term of
34:49 the sequence . So we're looking for a sub 10
34:53 . So let's plug intent into this equation . So
34:56 it's gonna be nine Times 10 -4 . nine times
35:00 10 is 90 90 -4 , is 86 . So
35:05 that is the 10th term of the sequence in part
35:09 A for part B . The 10th term is going
35:12 to be 157 -7 times 10 . seven times 10
35:18 is 70 1 , 57 minus 70 . It's gonna
35:23 be 87 . Now let's move on the part C
35:31 fined the sum of the 1st 10 terms . So
35:36 in order to find the sum we need to use
35:39 this formula ECE Ben is equal to the first term
35:45 plus the last term divided by two times the number
35:48 of terms . So if we want to find the
35:51 some of the 1st 10 terms , we need a
35:53 sub one which we know it's five A sub N
35:59 and is 10 . So that's eight of 10 .
36:02 The 10th term is 86 divided by two Times the
36:06 number of terms which is 10 five plus 86 is
36:11 91 91 divided by two . gives us an average
36:15 of 45.5 of the first and last number And then
36:19 times 10 We'll get a total sum of 455 .
36:24 So that is the sum of the 1st 10 terms
36:27 of this sequence . Now , for part B we're
36:33 gonna do the same thing , calculate acceptance . The
36:36 first term a someone is 150 . The 10th term
36:41 is 87 , divided by two times the number of
36:46 terms which is 10 , 1 50 plus 87 .
36:51 That's 2 37 Divided by two , that's 1 18.5
36:57 times 10 We get a sum of 1185 . So
37:05 now you know how to calculate the value of the
37:07 and turn and you also know how to find a
37:09 some of a serious number . Eight Find the sum
37:15 of the 1st 300 natural numbers . So how can
37:20 we do this ? The best thing we can do
37:22 right now is write a series . zero is not
37:26 a natural number , but one is . So if
37:28 we write a list one plus two plus three and
37:32 this is going to keep on going All the way
37:35 to 300 , So to find the sum of a
37:39 partial series , we need to use this equation as
37:42 seven is equal to a sub one Plus A 7/2
37:47 times end . Now let's write down what we know
37:52 , We know that ace of one . The first
37:54 term is one we know and is 300 . If
37:59 this is the first term , this is the second
38:00 term , this is the third term . This must
38:02 be the 300 term . So we know it is
38:07 300 and A seven or a sub 300 is 300
38:12 . So we have everything that we need to calculate
38:14 the sum of the 1st 300 terms . So it's
38:17 a sub one which is one plus a seven which
38:20 is 300 over to Times the number of terms ,
38:24 which is 300 . So it's going to be 301
38:28 divided by two times 300 And that's 40 5150 .
38:36 So that's how we can calculate the sum of the
38:39 1st 300 natural numbers in this series . Number nine
38:46 , Calculate the sum of all even numbers from 2
38:49 to 100 inclusive . So let's write a serious two
38:55 is even three is odd , so the next even
38:57 number is four And then six and then eight All
39:01 the way to 100 . So we have the first
39:05 term . The second term is for The third term
39:09 is 60 , 100 is likely to be the 50
39:13 of turn but let's confirm it . So what we
39:16 need to do is calculate end And make sure it's
39:18 50 and not 49 and 51 . So we're going
39:23 to use this equation to calculate the value event .
39:30 So a seven is 100 . Let's replace that with
39:34 100 . Ace of one is to the common difference
39:45 We can see 4 -2 is 2 6 -4 is
39:49 too . So the common difference is to in this
39:52 example and our goal is to solve for end .
39:56 So let's begin by subtracting both sides by two 100
40:02 -2 is 98 And this is going to equal two
40:06 times and -1 . Next we're going to divide both
40:10 sides by two 98 , divided by two is 49
40:16 . So we have 49 is equal to end -1
40:19 and then we're gonna add one to both sides .
40:23 So n is 49 plus one which is 50 .
40:28 So that means that 100 is indeed the 50th terms
40:32 . So we know that end is 50 . So
40:36 now we have everything that we need in order to
40:37 calculate the sum of the 1st 50 terms . So
40:41 let's begin by writing out the Formula 1st . So
40:48 the some of the 1st 50 of terms is going
40:50 to be the first term which is two plus a
40:53 some 50 . The last term which is 100 divided
40:56 by two times then which is 50 . So two
41:02 plus 100 . That's 102 divided by two . That's
41:04 51 51 times 50 Is 2550 . So that is
41:12 the sum Of all the even numbers from 2 to
41:15 100 inclusive . Try this one determine the sum of
41:19 all odd integers from 20 to 76 . 20 is
41:24 even but the next number 21 is odd And then
41:29 23 , 25 27 . All of that Are odd
41:34 numbers up until 75 . So a someone is 21
41:39 in this problem , the last number a sub N
41:46 is 75 . And we know the common differences too
41:52 because the numbers are increasing right . What we need
41:56 to calculate is the value event . Once we could
41:59 find end then we could find a some from 21-75
42:06 . So what is the value then ? So we
42:09 need to use the general formula for In a different
42:13 six sequence . So a sub n is 75 .
42:17 A sub one is 21 and the common difference is
42:21 too . So let's attract both sides by 21 ,
42:27 75 -21 . This is going to be 54 ,
42:35 Dividing both sides by two , 54 , divided by
42:41 two is 27 . So we get 27 is N
42:44 -1 and then we're going to add one to both
42:47 sides . So n is 28 . So a sub
42:52 28 is 75 , 75 is the 28th term in
42:58 the sequence . So now We need to find a
43:01 some of the 1st 28 terms . It's going to
43:04 be a someone the first term Plus the last term
43:08 or the 28th term , which is 75 divided by
43:11 two Times the number of terms , which is 28
43:21 , 21 plus 75 . That's 96 Divided by two
43:25 , that's 48 . So 48 is the average of
43:28 the first and the last term . So 48 times
43:31 28 , That's 1,344 . So that is the sum
43:38 of the 1st 28 terms .
Summarizer

DESCRIPTION:

This video provides a basic introduction into arithmetic sequences and series. It explains how to find the nth term of a sequence as well as how to find the sum of an arithmetic sequence. It also discusses how to distinguish a finite sequence from an infinite series. It also includes a few word problems.

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Arithmetic Sequences and Arithmetic Series - Basic Introduction is a free educational video by The Organic Chemistry Tutor.

This page not only allows students and teachers view Arithmetic Sequences and Arithmetic Series - Basic Introduction videos but also find engaging Sample Questions, Apps, Pins, Worksheets, Books related to the following topics.


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