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.

OVERVIEW:

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

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