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Grade 11 General Mathematics | Arithmetic Sequence and Arithmetic Series First Term (Term 1) Week 4 — Transcript

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0:00Hello everyone welcome to native man mat
0:03tutorial wherein you can find pre
0:05mathematics tagalog tutorial the content
0:08of this video lesson is general
0:10mathematics for grade 11 this is under
0:13term 1 week 4 of the strengthened senior
0:17high school curriculum
0:19for today's video lesson we are going to
0:22discuss arithmetic sequence and series
0:25but before we discuss this lesson if you
0:28are not yet subscribed to this channel.
0:30Please subscribe for more video lesson.
0:33So what are you waiting for? Let's
0:35begin.
0:37The learning competency of this lesson
0:40is find the end term of arithmetic and
0:43geometric series. And for the objectives
0:48at the end of the lesson, you will be
0:50able to number one define arithmetic
0:53sequence in series and its key
0:55components. Number two, differentiate
0:58arithmetic sequence from other
1:00sequences. Number three, find the end
1:03term of an arithmetic sequence. Number
1:06four, find the sum of arithmetic series.
1:09And number five, solve real life problem
1:12involving arithmetic sequence. and
1:15series.
1:17Okay, to define arithmetic sequence,
1:20let's do the activity. Consider the
1:23given set of numbers. Spot the
1:25differences of the sets of numbers.
1:28Okay, we have two sets of numbers. For
1:31set A, we have 2 4 6 8 and 10. And for
1:36set B, we have 1 2 4 11 51.
1:42So with these two sets of numbers we
1:45have the set a we have the so-called
1:49common difference. So why we have common
1:52difference here? If you're going to
1:54subtract the two terms or consecutive
1:57terms from right to left then there is a
2:01common difference. Meaning to say we
2:03have the same difference. So let's try.
2:07We have 10 - 8 is 2. We have 8 - 6 is 2.
2:12We have 6- 4 is 2. And 4- 2 is 2. So
2:16that two is called the common
2:18difference. How about on set B? Do we
2:22have a pattern? Do we have a sequence?
2:25Here we have 1 2 4 11 51. So here if you
2:31going to subtract 2- 1 is 1 but if you
2:35subtract here 4- 2 is 2. So there is no
2:39pattern here. For these two sets of
2:43numbers for set A we call this one as
2:46arithmetic sequence because there is a
2:49common difference. But for set B this is
2:52not arithmetic sequence. So meaning to
2:55say for arithmetic sequence we have
2:58common difference.
3:01Okay let's define what is arithmetic
3:05sequence. It is a number pattern in
3:08which every term after the first is
3:10obtained by adding a number to the
3:13preceding term. This means that in
3:15arithmetic sequence there is a common
3:18difference. So again nabanggit po yung
3:21common difference. So what is this
3:22again? Common difference in symbol D is
3:26the constant difference between the two
3:28consecutive terms in arithmetic
3:30sequence. It is also the number added to
3:33the term to get the next term. Okay? So
3:36iyung common difference is our basis so
3:40that we can say that the sequence is
3:42arithmetic sequence. So let's have
3:44examples here. For number one example we
3:47have 1 3 5 7. So if this sequence
3:53has a common difference d it is
3:57arithmetic sequence. So what is the
3:59common difference of number one
4:01sequence? The common difference d is
4:04equal to 2. Okay let's try. We are going
4:07to subtract the right term to the left
4:10term and so on and should have a common
4:13difference. So let's try. 7 - 5 is 2.
4:20And 3 - 1 is 2. So the common difference
4:24is 2. How about number two?
4:27We have 8 3 -2 -7. So what is the common
4:32difference? That is equal to -5. So if
4:36you're going to subtract from right to
4:38left then -7
4:41-2
4:44is equ -5. -2 - 3 is -5 and 3 - 8 is -5.
4:53Okay. How about number 3? 30
4:57and 10. Do we have common difference
4:59here? Yes. The common's difference is
5:02-10. How? 10 - 20 is -10. 20 - 30 is
5:10-10. Okay. So, mayro'n po siyang common
5:13difference. How about number four? -4,
5:18-2, 02.
5:21So meron po bang common difference?
5:23Okay. The common difference is equal to
5:262. So how? 2- 0 is 2. 0- -2 is 2 and -2
5:35-4
5:37is equ 2. Okay? So the common difference
5:40is 2. Now how about number five? It is a
5:43fraction.
5:45The sequence is 1/2 3/ 5/ and 7/
5:54going to subtract from right to left
5:57what is 7 2-
6:005 okay so the common difference there is
6:031. So let's try to subtract
6:067 / 2. Okay this one minus the this term
6:115 / 2. Kapag ang denominator
6:16are equal or the same then all you have
6:18to do is add or subtract the numerator.
6:22What is 7 - 5 that is equal to 2. So
6:26here we have 7 - 5 is 2 and then copy
6:30the denominator 2. So 2 / 2 that is
6:33equal to 1. So the common difference for
6:35number 5 is equal to 1.
6:45Okay. Let's try activity one. Let's see.
6:48How well do we know arithmetic sequence?
6:51Tell whether the set of numbers is
6:53arithmetic sequence or not. Okay, we
6:56have one, two, three and four. For
7:00number one, we have 2, 4 and 10. So, is
7:05this sequence or is this set of numbers
7:09arithmetic sequence or not? Okay, for
7:12number one, that is arithmetic sequence.
7:16Very good. How about number two? 5 10 20
7:2125. Okay. So what's your answer for
7:25number two sequence? Is it arithmetic or
7:29not? So the answer there is
7:33not arithmetic sequence. So why
7:35arithmetic sequence for number two? So
7:38if not arithmetic sequence meaning to
7:40say there is no common difference. So
7:42let's try to subtract. 25 - 20 is 5. 20
7:48- 10 is 10. So to say yung una the
7:53common difference ah the difference is 5
7:57at yung pangalawang set ang difference
7:59ay 10. So mean to say magkaiba sila ng
8:03difference so there is no common
8:05difference. So this is not arithmetic
8:06sequence. Okay? Unlike sa number one na
8:09meron silang common difference. So 10 -
8:138 is 2, 8 - 6 is 2 and so on. Okay. How
8:17about number three?
8:19-2 0 2 4 and 6. Is it arithmetic
8:24sequence or not?
8:28The correct answer is
8:30arithmetic sequence. So here there is a
8:33common difference. Okay. 6 - 4 is 2, 4 -
8:372 is 2, 2 - 0 is 2, and- -2 is 2 also.
8:42Okay. So we have common difference then
8:44that is arithmetic sequence. How about
8:46number four? 5 2 -1 -4
8:52arithmetic sequence or not? So that is
8:55still arithmetic sequence. Therefore, we
8:58have common difference here.
9:02To understand more about arithmetic
9:04sequence, we have this first term or a
9:08sub of one. So, what is this first term
9:11all about? The very first value in the
9:14sequence. It is the starting point from
9:17which other terms in the sequence are
9:20produced. So mean to you say ang first
9:23term is the first value in the sequence.
9:26So let's have this illustration.
9:29Terms of arithmetic sequence are a sub
9:331, a sub2 a sub 3, a sub 4, a sub 5 and
9:38until a sub n. So ano po yung a sub n?
9:42Kung hanggang saan po yung aabutin ng
9:45terms of the sequence can be a sub of
9:4850, a sub of 60 and so on. So kung
9:52hanggang saan yung arithmetic sequence.
9:55Okay. And a of one here that is called
9:58the first term. Okay. So here in our
10:02example for number one in arithmetic
10:04sequence we have 1 3 5 7 9 and so on.
10:10Okay? So here ang a 1 ay 1. Okay?
10:14Illustrate ko po dito. A 1 or first term
10:17is 1. Ace of 2 second term is 3. Ace of
10:213 or third term is 5. Ace of 4 is 7.
10:26this one a of 5 is 9 and so on. So here
10:31kung hanggang saan yung aabutin ng
10:33arithmetic sequence that is a sub n.
10:37Okay? So yung n here as I recall is the
10:40number of terms. And take note that the
10:45n is the specific term at the position n
10:49in the sequence. Okay, for example a 5
10:54okay is equ to 9. So 5 is n and the end
11:01term or a of 5 is 9. So yung position po
11:04ng 9 ay panglima sa ating sequence.
11:08Okay? So yun po yung ibig sabihin ng end
11:11term.
11:13To calculate the end term of an
11:16arithmetic sequence, use the formula.
11:19And what is the formula to calculate the
11:21end term? A sub n is equ to a 1 + open
11:27parenthesis n - 1 close parenthesis
11:31d. Okay? So yung close and open
11:34parenthesis ang operation nito sa d or
11:37common difference is multiplication.
11:40Okay? So here a sub n that is the end
11:44term. Okay. A sub 1 again the first term
11:48of the arithmetic sequence d as I recall
11:51that is the common difference and n the
11:54position of the term in the sequence or
11:58that is the number of terms. Okay,
12:02let's have an example applying the
12:06formula of n term. Number one, find the
12:10term of the arithmetic sequence 1 3 5 7
12:169 and so on. So what is the eight term
12:20of this arithmetic sequence? So first to
12:23answer this problem we're going to
12:26identify the given. So what are the
12:29given in the problem? Okay? So eight
12:32term or find the eight term. So 'yung a
12:35of 8 or pang 8 na term that is unknown
12:38kaya nilagyan ko po ng question mark.
12:41Ito po yung ating hahanapin. And from
12:45the formula we have a 1. So a of 1 is
12:49the first term. What is the first term
12:51in our sequence? That is 1. Kaya a 1 is
12:55equ 1. Next what else? Aside from a sub
12:59of 1 sa formula kailangan natin ng n at
13:02d. So here the D is 2. What is D again?
13:07Common difference. So what is the common
13:09difference of this sequence? Subtract
13:12from right to left. So 9 - 7 is 2, 7 - 5
13:17is 2, 5 - 3 is 2 and 3 - 1 is 2. So the
13:22common difference is equal to 2. And how
13:25about we need n in our formula? So n
13:30pang ilan daw po yung ating hahanapin.
13:33So pang 8 term. So n is equ to 8. So
13:37using this given we are going to
13:40calculate the term of the arithmetic
13:43sequence. Okay, from the given formula,
13:46okay, this one,
13:48we are going to substitute the given
13:51value here. For the ace of n, ito po
13:53'yung hahanapin natin pang 8. Okay? Pang
13:57po 'yung hahanapin natin pang 8 term.
13:59Kaya a sub 8 is equal to what is a sub
14:021? 1. So, we have one here. Then copy
14:06plus and then open parentheses. And what
14:09is our n from our given? N is 8 kaya 8.
14:13Then copy-us 1 this one then close
14:16parenthesis and then what is d or common
14:19difference that is equal to 2 okay so
14:22that's why we have times 2 here okay so
14:25now we are going to simplify this
14:27equation so ano po yung pwedeng
14:29i-simplify sa ating equation so
14:32magsu-subtract po muna tayo dito sa loob
14:34ng parenthesis 8 - 1. So this will
14:36become
14:38copy 1 then plus 8 - 1 is 7 and then
14:43times 2 and then before we add we
14:45multiply first. So what is 7 * 2 that is
14:49equal to 14 and then add 14 + 1 that is
14:53equ to 15. So mean to say the eighth
14:58term of this arithmetic sequence is 15.
15:01Okay. Kung duugtungan natin 'yung ating
15:03sequence, 9 + 2 kasi 2 po 'yung common
15:06difference. Add lang natin. 9 + 2 is 11.
15:1011 + 2 is 13, 13 + 2 is 15. So 'yung 15
15:14ba pangwalo? Ika-count po natin. Okay.
15:16Let's try to count. 1 2 3 4 5 6 7 8. So
15:25yung pangwalong term is equ to 15. Okay.
15:29So
15:31therefore 15 is the eight term of the
15:34arithmetic sequence.
15:37Number two, example. In the arithmetic
15:40sequence 3, 8 13 and so on. Find the
15:45value of n if a sub of n is equ to 68.
15:50Okay. So again to solve this problem
15:53applying the formula we need to identify
15:56the given. At ito po yung different
15:59given. So here a n is equ 68. 'Yan po
16:03yung hindi natin alam kung pang ilang
16:06term kasi hindi po natin alam kung ano
16:08po yung n the number of terms. So yun po
16:11yung ating hahanapin. Okay? A sub 1 in
16:15our given arithmetic sequence meron po
16:18dito yung pinakanguna 3. So a sub 1 is
16:21equ 3. And d or the common difference.
16:24So kailangan po natin ng d from the
16:26formula. So the common difference is 5.
16:30Tama po ba? So let's check.
16:3218 - 13 is 5. 13 - 8 is 5. 8 - 3 5. So
16:40the common difference is 5. Now n is the
16:44unknown. So how to calculate n from the
16:48given formula? Okay. So let's try. So
16:51ito po iyung ating formula. All you have
16:53to do is you're going to substitute all
16:56the given. Okay. So here a sub n is equ
17:0168. Substitute then follow equals a sub
17:051 or first term here is 3. This one.
17:08Then copy plus then open parenthesis and
17:12then n then -1. So yung n still n pa po
17:17siya kasi yun po yung ating iso-solve.
17:19Okay? Yun po yung ating unknown. and D
17:22the common difference is 5. So times 5.
17:25Okay. Here, how to start simplifying
17:29this equation? So, dito po ang gagawin
17:32po natin diyan we are going to
17:35simplify this open and close
17:37parentheses. Okay, simplify natin yung
17:40n- 1 5. By distributive property, 5 * n
17:46is 5n. and then min 5 * 1 or 1 5 that is
17:51equ to 5. Okay. So here the product of
17:55n- 1 5 is 5n- 5. Okay what's next? Wala
18:00na po yung close and open parenthesis.
18:03You are going to combine like terms. So
18:05how to combine? So this become
18:08combine 3 and -5 or what is 3- 5 that is
18:16-2. So yung 5n unahin na po natin. And
18:21then 3 - 5 is -2 that is -2. So we have
18:255n - 2. Okay. And then next po I combine
18:31-2 or -2 here and 68. So how we are
18:34going to move there? Okay. I-move natin
18:38yung -2 or -2 from right of equal sign
18:42sa left ng equal sign. So if you move,
18:46you're going to change the operation
18:48from minus to plus. So this will become
18:5268 + 2 is equ to 5n. Okay. So here add
18:5868 + 2 that is equ to 70. 70 equ 5n. So
19:05what's next? Ito na po yung ating
19:07equation. So we need n. Kailangan po
19:10natin yung n but we have 5n here. So
19:13we're going to move 5 to left of equal
19:16sign. So how to move five or remove five
19:18here? Divide by 5 to both side of the
19:21equation. We call this one as division
19:24property of equality or the
19:26multiplication property of equality and
19:30cancel 5. So again kung ano iyung
19:32idi-divide mo sa right of equal sign is
19:35also the number that you're going to
19:36divide sa left ng equal sign. Okay? So
19:39cancel 5. Then what is 70 / 5? That is
19:44equal to 14. Okay. And then we have n
19:47here. So 14 equ n is equ to 14. Okay.
19:53Mean to say
19:56that therefore 68 is the ah 68 in the is
20:02the 14th term of the arithmetic
20:05sequence. Okay? So yung 68 po ay pang 14
20:09na terms sa ating arithmetic sequence.
20:14For number 3 example the term of an
20:18arithmetic sequence is 13 and the term
20:22is 25.
20:24Find the following letter A question
20:27common difference letter B first term
20:31letter C the 30th term. Sumula po sa
20:35dalawang given ay hahanapin po natin
20:38yung tatlong questions na ito. Okay. So
20:42here ang hahanapin po natin yung una ay
20:45common difference or d. Okay? So lagay
20:47ko po dito common difference d is
20:50unknown. Second first term or a sub of
20:53one this one. And the third question is
20:56the 30th term or the a 30. Okay. Now, we
21:02all know from here ang end term or the
21:0530th term, the n is equ to 30. Pero
21:10paano po natin kaya kukunin yung common
21:12difference? Ito po yung ating given.
21:15Okay. A 4 or first term is equal to 13
21:20and the term is equal to 25. So paano
21:24natin kukunin yung common difference? '
21:26Ba sabi natin kanina that common
21:28difference is the result of subtracting
21:31two consecutive terms. So yung 4 at 8
21:35hindi po sila consecutive terms. So
21:37paano natin maso-solve yung common
21:39difference? Okay here. Ganito lang po
21:42yung ating gagawin. So here if 13 is the
21:46fourth term, okay, then 25 is the eighth
21:51term. So mula 13. Okay. Ito po yung a of
21:544 5 6 7 8. So may tatlo pong number ang
21:59pagitan ng 13 and 13 and 25. Now para
22:04ma-calculate natin yung common
22:06difference ay mag-set po tayo ng
22:08panibagong a sub 1 para ma-solve po
22:12natin yung d or common difference. So
22:14yung 13 ay gawin po nating a 1 or first
22:18term. Okay. Kaya kung ang 13 ay ace of
22:211, 'yung next number dito ay a 2. Next
22:24number dito ay ace of 3. The next number
22:26is a of 4. At ito pong 25 ay a 5.
22:32So here mula po sa concept na ito ay
22:35maco-compute po natin yung common
22:38difference. Okay? Here again i-set natin
22:41na a sub 1 is 13. At since panglima na
22:45dito sa ating set o sa ating sequence
22:48yung 25 that is a 5.
22:51Using the formula substitute the given.
22:55So a of n ito po yung panghuling number.
22:57Ang panghuling number po natin ay 25.
23:00Okay? So this one is equal to a 1 yung
23:04unang term that is 13. Kaya 13 po siya.
23:08Okay? This one. And then plus open
23:11parenthesis. What is our n? Ilan po yung
23:14ating terms dito? So to count 1 2 3 4
23:18and 5. Kaya there are five terms here.
23:20Kaya 5 then copy - 1 and then times d or
23:26common difference. So yung D po is
23:28unknown. Yun po yung ating hinahanap.
23:31Okay? So mula po dito sa ating equation
23:34notice here that the variable is only D.
23:38Kaya if you are going to simplify,
23:40maso-solve po natin yung common
23:42difference. Okay? How to simplify? We
23:45are going to subtract 5- 1 first. Okay?
23:48So this become
23:50copy 25 = 13 and then plus 4 5 - 1 is 4
23:58and then d kaya ito po ay 4D. Okay. So
24:03here what's next? I-move natin yung 13
24:06sa left ng equal sign. So this will
24:08become 25 - 13. So yung 13 po ay
24:12positive. If you move to left side of
24:15equal sign, change the sign from
24:17positive to negative. So yung negative
24:19will become minus. Okay? 25 - 13 is equ
24:244D. Okay? So here we can subtract 25 -
24:3013 that is equ to 12 is equ to 4D. Since
24:36D yung ating hinahanap, we're going to
24:38remove D by how? We're going to divide 4
24:44to both sides of equal sign. And we're
24:47going to cancel 4. Bakit po 4 angivide
24:50natin? Kasi 4 po yung kasama ng D or
24:53numerical coefficient of D. Okay. So,
24:56divide by 4 to both side and can then
24:59cancel 4. Divide 13 / 4 is equ to 3. So
25:053 is equ to D. So therefore here the
25:09common difference is 3. To answer in
25:12sentence. Therefore, the common
25:15difference is 3. So, na-solve na po
25:17natin yung first question that D is equ
25:21to 3. Okay, next, we are going to
25:24calculate the second question, first
25:26term or a sub of 1. So, paano naman kaya
25:30natin maso-solve yung a? Okay, ito po
25:33ulit yung ating bases. 13 to 25 here.
25:36So, itaas ko po dito. So, 13 then to 25.
25:41So again 'yung 13 po ay ito po 'yung a
25:454. So balik na po tayo sa kanyang given.
25:48A 4 is equ 13 and a 8 = 25. Okay? So
25:54here
25:56yung 13 ay a of 4. Tama po ba? So
25:58meaning to say kung pang-apat yung 13
26:01may tatlong numbers before 13. Okay.
26:04Kaya 13 is a 4. So we have a 3, a 2, and
26:11a of 1. First term, second term and
26:13third term at pang-apat ay 13. So mean
26:16to say ang 25 po yan po yung pang walo.
26:20The 8 term is equal to 25. Now paano po
26:25natin makukuha yung a sub 1 dito? First
26:27term of this sequence. So para
26:31ma-compute natin yung a one, dito po
26:33tayo magfo-focus. Okay? And using the
26:36formula, we are going to substitute the
26:39given. What is a sub n? Yung panghuling
26:42number mula sa blank na ito hanggang sa
26:44dulo. Yung panghuling number ay 13.
26:47Okay, substitute. We have 13 is the last
26:51uh term here is equal to what is a sub
26:54of 1? Unknown po. Wala pa pong a sub 1
26:57dito. Kaya a sub of 1 and then plus what
27:00is n? N is the number of terms. So ilang
27:03terms po lahat ito? Mula po a 1 hanggang
27:06a sub of 4 we have four terms. So 1 2 3
27:094. So we have four terms kaya 4 - 1
27:13times what is the common difference? D.
27:16So na-solve na po natin d is equ to 3.
27:19So times 3 here. Okay. Now simplify the
27:24equation. Notice here a sub 1 lang po
27:27yung variable dito. Mean to say kapag
27:29sinimplify natin maka-calculate natin
27:32yung a one or first term. So ito po muna
27:35ang simplify natin. So this will become
27:3813 copy is equ to a sub 1 and then plus.
27:43Okay, subtract first this one nasa loob
27:45ng parenthesis and then multiply by 3.
27:484- 1 is 3. Then multiply it by 3. 3 * 3
27:53that is equ 9. So pos 9 that will become
27:56+ 9. Okay. What's next? Yung 9 i-move
28:00natin sa 13. Okay? Sa left ng equal
28:03side. Okay. This will become so + 9. If
28:06you move that to left of equal sign,
28:08change the sign or change the operation
28:11from plus to minus. Kaya ito po ay 13 -
28:149 equ to ang matitira po dito ay a 1. So
28:18what is 13 - 9? That is equ to 4. So
28:23here 4 is a 1 or the first term is 4.
28:29Okay? To answer in sentence. Therefore,
28:31the first term is 4. Okay, lagay na po
28:34natin that a 1 here is equal to 4.
28:40Okay, last we are going to calculate the
28:43third question the 30th term or the a of
28:4730. So again, using the formula of the
28:50end term, a n a 1 + n - 1 d. So here ang
28:57a a sub n yan po yung hahanapin natin.
29:00Yan po yung a sub 30 or pang 30 term. So
29:04here the first term is 3 ah sorry 4. A
29:08sub 1 is 4 here plus n. So ilan po yung
29:13number of terms? Since yung a n a sub of
29:1530 po yung n natin kaya 30. Then copy- 1
29:21d what is common difference 3 so 3 here
29:24so simplify how to simplify subtract
29:27first 30 - 1 that is equ to 29 so 4 + 29
29:34* 3. So multiply and then add. So this
29:38will become
29:4029 * 3 is 87 and then 4 + 87 that is equ
29:45to 91. So here therefore the 30th term
29:49is 91. So lagay po natin okay this one.
29:56Okay we are going to apply the concept
29:58of arithmetic sequence in solving real
30:01life problems. For problem number one,
30:05money is taking a road trip and plans to
30:08increase his daily driving distance. He
30:12drives 100 miles on the first day and
30:15increases his distance by 25 miles each
30:18day.
30:19If he continues this plan for a week,
30:23how far will he drive on the last day?
30:27Okay. So, how we going to calculate what
30:30is asked in the problem? So, the problem
30:33asked, "How far will he drive on the
30:35last day?" Bakit po meron siyang last
30:38day? Okay. So, yung plan po niya ay for
30:41the week. Here, the plan is for the
30:44week. So how many days are there in the
30:47week? Okay. So there are 7 days a week.
30:53Okay? So identify the given. So yung
30:56pangse yun po yung ating hahanapin. So a
30:59of 7 ay unknown. Okay? That is the last
31:03day, the seventh day of the week. Then
31:07the first term here is 100. So bakit
31:09100? He drives 100 mil on the first day.
31:13So yung first term natin ay 100. Okay?
31:17And what is the common difference? Why
31:1925 is the common difference? So sabi
31:22niya, "It increases his daily distance
31:27by 25 miles." So bawat araw magdadagdag
31:32siya ng 25 miles na ida-drive niya.
31:35Okay? Kaya ang common difference po
31:37natin kasi ang pagitan po ng terms ay 25
31:41miles. Okay? 25 and then n of terms ay
31:467. Okay? So applying the formula of
31:49sequence, we have this formula and
31:52substitute the given. So a sub of n ang
31:55hahanapin natin ay pang seven term the
31:57last day of the week. So a sub of 7 is
32:00equ to the first term natin a sub 1 is
32:03100. Okay. Plus n - 1. Our n is 7 kasi
32:09pang 7 na araw. So 7 then copy -1 times
32:13the common difference d is 25. Okay.
32:17Subtract and then multiply. So this will
32:19become
32:217 - 1 is 6 then times 25. So what is 6 *
32:2525? So this will become 150.
32:30Okay. So we have here 100 + 150 that is
32:35equ to 250.
32:37So here therefore money will drive 250
32:42mil on the last day or that is the 7th
32:46day. Okay.
32:49Okay. Another activity to master the
32:52lesson. We have activity 2. Choose the
32:55letter of the best answer. We have 1 2 3
33:00and 4 questions here. Okay. For number
33:02one, which formula is used to find the
33:06end term of an arithmetic sequence? We
33:09have A, B, C, and D. Okay, you can post
33:15this video and try to answer. So, what's
33:18your answer for number one?
33:21I will give you 3 seconds. The answer
33:24there is letter C. How about number two?
33:29What is the common difference of the
33:31arithmetic sequence 6 10 14
33:36and so on? So we have A2, B4, C5 and D3.
33:42So for number two, it ask the common
33:45difference. So what's your answer? I
33:47will give you 5 seconds.
33:53Okay. The correct answer for number two
33:56is letter B. So the common difference is
33:594. How about number three? Which of the
34:02following is the third term of the
34:04arithmetic sequence? 1 6 11 16. Simula
34:10sa ating arithmetic sequence, what is
34:12the third term? Okay, I will give you 3
34:16seconds to have an answer.
34:21The correct answer for number 3 is 11.
34:25So 11 po yung pang third term. So first
34:28term, second term, and third term.
34:30Number four. What is the what is the age
34:33of n of the arithmetic sequence 4 7 10
34:3913?
34:40Okay. So we have a b c and d. 3 seconds
34:45to answer.
34:51Okay. The correct answer there is letter
34:53D 13.
34:57How about this activity number 3? Solve
35:00the problem. We have one, two, three
35:03problems here. Okay. So, I'll give you
35:082 minutes or 1 minute to answer these
35:11three problems.
35:13Try to post this video and calculate for
35:18answers number one, 2, and 3. And then
35:21later you can continue watching this
35:23video and try to see if your answer is
35:26correct. Okay. So for number one, the
35:30answer is
35:3324 or the 8 term is 24.
35:37For number two, n is equ 23. And for
35:41number 3, a sub of 1 is equ to 16.
35:47Okay. So let us recall the definition of
35:49arithmetic sequence by having this
35:52activity.
35:54To define arithmetic series, let's do
35:57the activity. Look and analyze the given
36:00sequence and spot the differences. Okay.
36:03So here set A and set B. For set A we
36:08have 2 4 6 8 and 10. For set B 2 + 4 + 6
36:15+ 8 + 10 is equ 30. So what is the
36:19difference between the two sets? Okay.
36:23For set A that is a sequence
36:26and for set B yan po yung sum of
36:30arithmetic sequence. So bakit po sum of
36:33arithmetic sequence for set B? We are
36:36going to add the terms in a sequence.
36:39Here sa atin pong set A just a set or
36:43just an arithmetic sequence. Okay. So
36:46here the sum of arithmetic sequence is
36:50called arithmetic series. So mean to say
36:53pag sinabi nating arithmetic series
36:56understood that you are going to find
36:59the sum of arithmetic sequence. Okay?
37:04Arithmetic series is the sum of terms of
37:07an arithmetic sequence. It is denoted by
37:11sn of n. So what is S? Ito po ay sum of
37:15n terms in the arithmetic series. Okay.
37:19So here is the illustration of
37:23arithmetic series. S n or sum of the
37:28terms is equal to a sub 1 + a sub2 + a
37:32sub 3 + and so on. Hanggang a sub of n.
37:37Ang a nto yung panghuling terms. Okay.
37:41So here pag sinabi nating arithmetic
37:44series all you have to do is add all the
37:47terms in the arithmetic sequence. Okay?
37:51So here there are two formula that can
37:53be used in
37:56solving the arithmetic series. Okay, for
37:59the first formula, if the last end term
38:03or a of n is given, okay, so ito po yung
38:06formula na gagamitin natin kapag yung
38:09last term ay given. A sub n/ 2 a sub 1 +
38:16a sub n. Okay. How about if the last
38:19term a sub n is not given? So hindi po
38:23given yung panghuling term or panglast
38:25term. So this is the formula that we are
38:27going to use. A sub of n is equ n/ 2.
38:32Then we have here times 2a 1 + open
38:37parenthesis n - 1 close parenthesis
38:39times d. So ano po yung ibig sabihin ng
38:41formula? So we discuss this later on in
38:45our examples. Okay. So what does it mean
38:48in our formula? A n the sum of the first
38:52n terms
38:54and the number of terms. A sub of 1
38:57again. Kailangan po natin sa
38:59pagka-calculate ang a sub one first
39:01term. A sub of n the last term. Meron po
39:04tayong a sub of n dito. At here wala
39:07pong a sub n. Okay. And for D that is
39:11the common difference. So here in our
39:14first formula, no need common difference
39:17but in our second formula we need d or
39:20common difference.
39:23Okay we are going to apply the formula
39:26in solving arithmetic series. Example
39:30number one, find the sum of the first 25
39:33terms of an arithmetic sequence 18, 23,
39:3828, 33 and so on. Okay? So mula po sa
39:43problem na ito, we are going to
39:45calculate the sum of the first 25 terms.
39:48Okay? To calculate, we're going to
39:50decide which formula here we are going
39:54to use. So para makapag-decide tayo kung
39:57ano iyung gagamitin nating formula,
39:59identify the given. Okay? So the given
40:02there is or r. We have s n. Ito po iyung
40:07sum of 25 terms that is unknown kaya
40:09question mark. And the number of terms.
40:12So how many terms are there? 25 terms.
40:16Kasi sum of the first 25 terms. Okay? So
40:19our n or number of terms is 25. And a
40:23sub of 1 18. Saan po nakuha iyung a 1.1?
40:27Dito po sa ating sequence. So here in
40:30our sequence 18 is the first term or a
40:33sub 1 and d is 5 common difference. So
40:37mula din po dito sa ating sequence
40:40makukuha natin yung d or common
40:43difference by subtracting consecutive
40:46terms from right to left. 33 - 28 is 5.
40:50Correct. 28 - 23 is equ 5 also and 23-
40:5618 also. So the common difference is
40:59equal to 5. Now we are going to decide
41:03from this given the formula that we are
41:06going to use. Is it the first formula or
41:08the second formula? So sa atin pong
41:11given, wala pong last term or a sub n.
41:14So hindi natin pupwedeng gamitin yung
41:16first formula kasi kailangan niya ng
41:19last term. Okay, we use the second
41:22formula. So this will become
41:25we have s n is equ n/ 2a
41:291 + n- 1 d here substitute the given. So
41:34this will become
41:36s n or sum of n terms here the sum of 25
41:40terms kaya s of 25 is equal to what is n
41:44again 25 terms. So 25 then divide by 2
41:47copy 2 here. Then we have this one. 2 is
41:52constant. A sub of 1 is 18. So 2 * a sub
41:561 po ang ibig sabihin nito kaya 2 * 18.
41:59Okay then copy plus what is n again n is
42:03equ to 25 then - 1 kaya 25 - 1 d or
42:09common difference which is 5 here. So
42:12times 5. Now how we simplify this
42:16equation? Okay. Ang una pong
42:18isi-simplify ay ito pong nasa loob ng
42:22parenthesis 25- 1. Okay. So this will
42:26become here.
42:29Okay. So dito po pwede na po nating
42:31isabay sa 25 - 1 = 24. Ito pong 2 * 18.
42:36Okay? 2 * 18 is equ 36. Okay? So copy
42:41lang muna natin yung 25/ 2. This will
42:44become next 2 18 is 36. and then plus 25
42:49- 1 is 24 times the common difference 5
42:52kaya 24 * 5. So bago tayo mag-add ah
42:57multiply po muna tayo. So this become
43:00just copy 25 / 2 then times 36 + 25 ah
43:0624 * 5 is 120
43:10okay and then add 36 + 120 that is equ
43:14to 156.
43:17Okay. So here pa na po natin 'yan
43:19isi-simplify. Mayon po tayong fraction
43:21before. So ang gagawin lang po natin
43:23diyan we are going to multiply 25 * 156.
43:27So multiply 25 or yung numerator at yung
43:31whole number dito. Okay? So this will
43:33become here. Dito ko ilalagay. 25 * 156
43:38is 3,900.
43:40and then copy the denominator 2. Okay,
43:42so this one and divide 3,900/
43:462 that is equal to 1,950.
43:51So the sum of the first 25 terms is
43:54equal to 1950. Okay?
43:58So in sentence we have here the sum of
44:01the first 25 terms is 1,950.
44:07How about this example number two? Find
44:10the sum of the arithmetic sequence with
44:1410 terms 3 10 17 until 66. Okay? So ito
44:21pong ating sequence ay mayroong 10
44:24terms. So 66 is the last term. Ito po
44:27yung pang 10. Okay. Now we're going to
44:30find the sum. So identified the given so
44:34that we can decide which formula we are
44:37going to use here. Okay? Ito po yung
44:39ating given s 10. Okay. So find the sum
44:43of the arithmetic sequence with 10
44:46terms. Kaya sum of 10 terms po ito. That
44:48is our unknown. And then n. So 10 terms
44:52po lahat. Kaya ang number of terms
44:54number of terms natin or n is equal to
44:5610. And the first term mula po dito sa
44:58ating sequence the first term is 3. So a
45:011 is equ 3. And then last a n yung
45:05panghuling term. Ito po yung panghuling
45:07term. 66. Now do we have last term here?
45:12Yes we have last term which is 66. Now
45:15we can use the first formula. Okay to
45:18use the first formula, okay this one
45:21substitute the given. A sub of n is the
45:25sum of 10 terms. Kaya substitute s 10 is
45:29equ to n number of terms is 10. So 10 /
45:322 times a sub 1 is 3. The first term
45:36plus the last term a sub of n is equ to
45:4066. Okay? Mas madali po yung computation
45:43niya. Then next we just add first 3 + 66
45:47that is equ to 69 and then divide kasi
45:51pwede pong i-divide whole number 10/ 2
45:54is equ 5. Okay and multiply 5 * 69 that
46:00is equ to 345.
46:03So the sum of the first 10 terms of the
46:08sequence is 345.
46:13For example number kuya is a tricycle
46:18driver in the newview of Aurora. On his
46:23first day he earns Php2.
46:26As more tourists visit throughout the
46:29week, his earnings increased by Php1
46:34day. If this pattern continues for 15
46:37days, how much will kuya earn in total?
46:42Okay, so this is problem solving about
46:45arithmetic series. Okay. So here what
46:49formula that we are going to use to
46:52solve this problem? Of course we are
46:54going to identify the given first. Okay,
46:58we have we are going to calculate the
47:00sum of the first 15 days kaya
47:04a s of 15 the sum of 15 terms. Okay.
47:09Then next,
47:11what is n? N is equ to 15 kasi 15 days.
47:15And the first term a 1 is 200. Yung 200
47:19ay nag-umpisa siyang nag-earn ng Php200
47:22kaya first term natin ay 200. And the
47:26common difference is 100. So how you get
47:28the common difference as 100? Ganito
47:32lang po 'yun. Okay? It says that as more
47:35tourist visit throughout the week, his
47:38earnings increase by 100 each day. Ibig
47:41sabihin bawat araw ay 100 po yung
47:45dagdag. Ibig sabihin ang common
47:48difference o ang pagitan ng bawat ah
47:51terms ay 100. Okay. So in our given,
47:58what formula that we are going to use?
48:00Okay. May term po ba? Wala pong last
48:03term. Kaya ito po yung ating gagamitin,
48:06second formula. Okay. So here in our
48:09second formula, substitute the given. So
48:11this become
48:14s n or sum of n. So sum of 15 terms 15
48:19po lahat is equal to what is n again 15
48:22/ 2 then times 2 a 1 is 200 so 2 * 200
48:30then plus n - 1 n is 15 kaya here 15 - 1
48:36d the common difference is 100 so 100
48:40here so again we can simplify 2
48:4515- 1. So this will become just copy s
48:4915 = 15 / 2 then times 2 is 400 + 15- 1
48:59is 14 and then times 100 so multiply and
49:04then add so this will become
49:0714 100 is 1400 and then we going to add
49:11that by 400 so this become we have here
49:16400 + 1,400
49:19is 1,800.
49:22So how we going to simplify this one
49:23with fraction? So pwede pong i-divide
49:26and then multiply by 1,800.
49:29Or the other way naman you can multiply
49:3315 1,800
49:36then divid by 2. Yan po yyung aking
49:38ginawa. So 15 * 1,800 is 27,000 then
49:43divid by 2 kasi ang denominator ng
49:46fraction here is 2 kaya divide by 2. And
49:48if you divide then the quotient is
49:5213,500.
49:54So the sum of 15 terms is equal to
50:0013,500.
50:02So ano po yung 10,000 13,500 from the
50:05problem? Ito po yung kita ni kuya maki.
50:08So therefore, kuya will earn a total of
50:12Php13,500
50:14in 15 days.
50:18Okay, let's try to apply arithmetic
50:20series in some real life problems. For
50:24problem number one, Mang Tomas, a driver
50:27or a gyney driver in Baguio starts with
50:30a fair of Php20
50:32on his first trip of the day. For its
50:36succeeding trip, he earns Php5 more than
50:40the previous trip. How much will he earn
50:43in total after 10 trips? Okay, so here
50:47what formula that we are going to use?
50:50Now, we're going to illustrate the
50:51problem. So si Mang Tomas daw po ay
50:54driver sa Baguio. Okay? At yung unang
50:58kita po niya ay Php20 sa first trip.
51:03Kaya ang first term po natin ay Php20.
51:06Now for its succeeding trip he earns
51:10Php5 more than the previous trip. Okay?
51:13So magdadagdag tayo ng Php5 for every
51:16trip. So yung first trip niya ay 20. So
51:19second trip po niya ay + 5 kaya 25.
51:22Third trip po niya plus 5. So 25 + 5 is
51:2530. And fourth trip po niya plus 5 ulit.
51:29So 30 + 5 is 35. And so on. Ganun po
51:32yung mangyayari. Now the question asked
51:35how much will he earn in total after 10
51:38trips? So in 10 trips, how much is the
51:43uh amount uh earned by manghas? Okay.
51:49Identify the given. Sum of 10 terms is
51:53unknown. N is 10. First term is 20. This
51:57one. And common difference is 5. Okay.
52:00Five po yung common difference. Now
52:02using the formula, okay, substitute the
52:05given. So ang gagamitin po natin dito ay
52:08iyung second formula since wala pong
52:09last term. Okay? So s 10 is equ to n is
52:1310/ 2. So n / 2
52:17* a sub 1 is 20. Kaya 2 * 20 then plus n
52:23number of terms is 10. So n- 1 so 10- 1
52:27and then common difference. So times d5.
52:30Okay. So, we're going to simplify this
52:33one. Pwede na rin pong isabay itong 10 /
52:362 kasi whole number 'yung magiging
52:38lalabas. So, this will become can be 10
52:41/ 2 is 5. Pwede na po 'yan. Okay. Then
52:45times 2 * 20 is 40. Then plus 10 - 1 is
52:519 and then times 5. Kaya 9 * 5. So,
52:54multiply first, then add.
52:57So this will become 9 * 5 is 45 and then
53:01+ 40. So add 40 + 45 here is 85 and then
53:08multiply it by 5. So 5 * 85 that is equ
53:11to 425.
53:14So here therefore Mang Thas earn Php425
53:21in 10 trips.
53:25Okay, let's have this activity number
53:27four. Choose the letter of the best
53:29answer. We have number one. What is the
53:32main difference between a sequence and a
53:35series? Okay, try to read and answer in
53:39your own. So later I will give the
53:42correct answer and also for number two,
53:45I will give you one minute to answer
53:48these two questions or you can post this
53:52video lesson and then later ay i-play po
53:56ninyo to reveal what is the correct
53:58answer for number one and number two.
54:01Okay? Try to post and answer.
54:05So let's answer number one and number
54:06two. For number one, the correct answer
54:09is B. And for number two, the correct
54:11answer is letter C.
54:15Okay, another number three and number
54:17four question. Okay, so again, try to
54:21read number three and number four and
54:23then answer in your own. I will give you
54:26time again to answer. Or pwede po
54:29ninyong i-post again yung video, try to
54:31answer and later play and then check if
54:35your answer is correct or wrong. Okay.
54:38So for number three, the correct answer
54:41is C. And for number four, the correct
54:44answer is letter D.
54:47Okay. To end up the discussion, I will
54:50give you exercises. Okay. Or this is
54:53called practice problems. So if you got
54:56the correct answer here, well,
54:58congratulation. If not, then you can
55:02study again our discussion. Okay, pwede
55:05lang po ninyong i-replay ito pong ating
55:07video lesson. Or baka naman po pwede
55:10nyong i-post 'yung video na ito, try to
55:12answer one, two, and 3. And if you have
55:16your answers, then try to type your
55:18answer in the comment section below this
55:21video lesson. Okay? Hihintayin ko po
55:24iyung inyong sagot.
55:26Thanks for watching guys. See you on my
55:29next discussion about geometric
55:31sequence. y

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