Quarter I - Module 4 Finding The Sum Of The Terms Of A .
10MathematicsQuarter I - Module 4Finding the Sum of the Terms ofa Given Arithmetic Sequence
Mathematics – Grade 10Alternative Delivery ModeQuarter I – Module 4: Finding the Sum of the Terms of a Given ArithmeticSequenceFirst Edition, 2020Republic Act 8293, section 176 states that: No copyright shall subsist in anywork of the Government of the Philippines. However, prior approval of the governmentagency or office wherein the work is created shall be necessary for exploitation of suchwork for profit. Such agency or office may, among other things, impose as a conditionthe payment of royalties.Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names,trademarks, etc.) included in this module are owned by their respective copyrightholders. Every effort has been exerted to locate and seek permission to use thesematerials from their respective copyright owners. The publisher and authors do notrepresent nor claim ownership over them.Published by the Department of EducationSecretary: Leonor Magtolis BrionesUndersecretary: Diosdado M. San AntonioDevelopment Team of the ModuleWriter’s Name:Evelyn N. BallatongEditor’s Name:Laila B. Kiw-isenReviewer’s Name: Bryan A. HidalgoManagement Team:May B. Eclar, PhDBenedicta B. GamateroCarmel F. MerisMarciana M. Aydinan, PhDEthielyn E. TaquedEdgar H. MadlaingLydia I. BelingonPrinted in the Philippines by:Department of Education – Cordillera Administrative RegionOffice Address:Telefax:E-mail Address:Wangal, La Trinidad, Benguet(074) 422-4074car@deped.gov.ph
10MathematicsQuarter 1 - Module 4Finds the Sum of the Terms of aGiven Arithmetic SequenceM10AL – Ic - 2
Introductory MessageThis module was collaboratively designed, developed and reviewed byeducators both from public and private institutions to assist you, the teacheror facilitator in helping the learners meet the standards set by the K to 12Curriculum while overcoming their personal, social, and economicconstraints in schooling.This is a part of the fourth learning competency in our Mathematics 10curriculum standard hence mastery of the skills is significant to have asmooth progress in the succeeding lessons.For the facilitator:Hi. As the facilitator of this module, kindly orient the learner on how to goabout in reading and answering this learning material. Please be patient andencourage the learner to complete this module. Do not forget to remind thelearner to use separate sheets in answering all the activities found in thismodule.For the learner:Hello learner. I hope you are ready to progress in your Grade 10 Mathematicsby accomplishing this learning module. This is designed to provide you withinteractive tasks to further develop the desired learning competencies onfinding the sum of the terms of an arithmetic sequence. This module isespecially crafted for you to be able to cope with the current lessons taken byyour classmates. Please read completely the written texts and follow theinstructions carefully so that you will be able to get the most of this learningmaterial. We hope that you will enjoy learning.Here is a guide on the parts of the learning modules which you need tounderstand as you progress in reading and analyzing its content.ICONLABELDETAILWhat I need to knowThis will give you an idea of the skills orcompetencies you are expected to learn inthe module.What I knowThis part includes an activity that aimsto check what you already know aboutthe lesson to take. If you get all theanswers correct (100%), you may decideto skip this module.What’s InThis is a brief drill or review to help youlink the current lesson with the previousone.ii
What’s NewIn this portion, the new lesson will beintroduced to you in various ways suchas a story, a song, a poem, a problemopener, an activity or a situation.What Is ItThis section provides a brief discussion ofthe lesson.This aims to help youdiscover and understand new conceptsand skills.What’s MoreThis comprises activities for independentpractice to solidify your understandingand skills of the topic. You may checkthe answers to the exercises using theAnswer Key at the end of the module.What I have LearnedThis includes questions or blanksentence/paragraph to be filled in toprocess what you learned from thelesson.What I Can DoAssessmentThis section provides an activity whichwill help you transfer your newknowledge or skill into real life situationsor concerns.This is a task which aims to evaluate yourlevel of mastery in achieving the learningcompetency.Additional ActivitiesIn this portion, another activity will begiven to you to enrich your knowledge orskill of the lesson learned. This alsotends retention of learned concepts.Answer KeyThis contains answers to all activities inthe module.At the end of this module you will also find:ReferencesThis is a list of all sources used indeveloping this module.iii
The following are some reminders in using this module:1. Use the module with care. Do not put unnecessary mark/s on anypart of the module. Use a separate sheet of paper in answering theexercises.2. Don’t forget to answer What I Know before moving on to the otheractivities included in the module.3. Read the instruction carefully before doing each task.4. Observe honesty and integrity in doing the tasks and checking youranswers.5. Finish the task at hand before proceeding to the next.6. Return this module to your teacher/facilitator once you are throughwith it.If you encounter any difficulty in answering the tasks in this module, do nothesitate to consult your teacher or facilitator. Always bear in mind that youare not alone.We hope that though this material, you will experience meaningful learningand gain deep understanding of the relevant competencies. You can do it!iv
WHAT I NEED TO KNOWThis module was designed and written with you in mind. It is here to help youfind the sum of the terms of an arithmetic sequence. The scope of this modulepermits it to be used in many different learning situations. The lessons arearranged to follow the standard sequence of the course but the pacing inwhich you read and answer this module will depend on your ability.After going through this module, you are expected to be able to demonstrateknowledge and skill related to sequences and apply these in solving problems.Specifically, you should be able to:a)b)c)define arithmetic series,find the sum of the firs 𝑛 terms of a given arithmetic sequence, andsolve word problems involving arithmetic series.WHAT I KNOWFind out how much you already know about the topics in this module. Choosethe letter of the best answer. Take note of the items that you were not able toanswer correctly and find the right answer as you go through this module.Write the chosen letter on a separate sheet of paper.1.It is the sum of the terms of a sequence.A) meanB) sequenceC) nth termD) series2.Find the sum of the first ten terms of the arithmetic sequence 4, 10, 16, A) 310B) 430C) 410D) 3903.Find the sum of the first 25 terms of the arithmetic sequence 17, 22,27,32, A) 1925B) 1195C) 1655D) 18951
4.The sum of the first 10 terms of an arithmetic sequence is 530. What isthe first term if the last term is 80 and the common difference is 2.A) 64B) 54C) 34D) 445.The third term of an arithmetic sequence is 12 and the seventh term is8. What is the sum of the first 10 terms?A) 5B) 8C) 11D) 156.Find the sum of the first 50 terms of the arithmetic sequence if the firstterm is 21 and the twentieth term is 154.A) 9635B) 9765C) 9265D) 96257.Find the sum of the first eighteen terms of the arithmetic sequence whosenth term is 𝑎𝑛 15 8𝑛.A) 1438B) 1638C) 1836D) 17838.The first term of an arithmetic sequence is 5, the last term is 45 and thesum is 275. Find the number of terms.A) 13B) 10C) 12D) 119.If the first n terms of the arithmetic sequence 20, 18, 16,. are added,how many of these terms will be added to get a sum of -100?A) 35B) 25C) 15D) 3010. A yaya receives a starting annual salary of Php 36,000.00 with a yearlyincrease of Php 3,000.00. What is her total income for 8 years?A) Php 315,000.00B) Php 372,000.00C) Php 432,000.00D) Php 495,000.0011. Jane was saving for a pair of shoes. From her weekly allowance, she wasable to save Php 10.00 on the first week, Php 13.00 on the second,Php 16.00 on the third week, and so on. If she continued saving in thispattern and made 52 deposits, how much did Jane save?A) Php 3,984.00B) Php 4,568.00C) Php 4,498.00D) Php 5,678.0012. Mary gets a starting monthly salary of Php 6,000.00 and an increase ofPhp 600.00 annually. How much income did she receive for the first threeyears?A) Php 276,300.00B) Php 237, 600.00C) Php 637, 300.00D) Php 673, 200.002
13. Mirasol saved Php10.00 on the first day of January, Php12.00 on thesecond day, Php14.00 on the third day, and so on, up to the last day ofthe month. How much did Mirasol save at the end of January?A) Php 2,710.00B) Php 2,170.00C) Php 1,240.00D) Php 1,420.0014. Mrs. De la Cruz started her business with an income of Php 125,000.00for the first year and an increase of Php 5,000.00 yearly. How much isthe total income of Mrs. De la Cruz for 8 years since she started herbusiness?A) Php 1,104,000.00B) Php 1,410,000.00C) Php 1,140,000.00D) Php 2,140,000.0015. A hall has 30 rows. Each successive row contains one additional seat. Ifthe first row has 25 seats, how many seats are in the hall?A) 1,185B) 1,815C) 1,970D) 1,780Lesson1Finding the sum of the firstn terms of a givenarithmetic sequenceWHAT’S INIn the previous module, it was discussed that to find the nth term of a givenarithmetic sequence, the formulaan a1 d(n – 1) can be used.This module will be discussing how to find the sum of the first n terms in anarithmetic sequence.3
For example, how do we compute the sum of all the terms of each of thefollowing sequences?a) 1, 2, 3, . . . , 100b) 5, 10, 15, 20, . . . , 50c) 5, 2, 1, 4, . . . , 31Adding manually the terms of a sequence is manageable when there are onlyfew terms in the sequence. However, if the sequence involves numerous terms,then it is no longer practical to be adding the terms manually. It is a tediouswork to do. Thus, this module will present to you a formula that will makethe computation easier and faster.WHAT’S NEWTo let you experience getting the sum of the terms in a sequence manually,do the following.1. Find the sum of the first 20 natural numbers.Solution:a. By listing all the natural numbers from 1 to 20 and adding them, wehave:1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 210b. Thus, the sum of the first 20 natural numbers is 210.2. Find the sum of all the terms of the sequence: 5, 10, 15, 20, , 50.Solution:a. By listing all the terms of the sequence and adding them, we have:5 10 15 20 25 30 35 40 45 50 275b. Thus, the sum of the terms of the sequence is 275.4
3. Find the sum 5, 2 , 1, 4, , 31.Solution:a. By listing all the terms of the sequence and adding them, we have: 5 ( 2) 1 4 7 10 13 16 19 22 25 28 31 169b. Thus, the sum of the terms of the sequence is 169.In doing this kind of solution, it is very challenging specially if you are dealingwith a sequence that has many terms. For example, finding the sum of theterms of the sequence: 1, 2, 3, . . . , 10,000. There are 10,000 terms to beadded one by one to get their sum.To derive a formula to be used in finding the sum of the terms of an arithmeticsequence, consider the following illustration:The terms of an arithmetic sequence with common difference, 𝑑, areFirst termSecond termThird termFourth term𝑛th𝑎1𝑎1 𝑑𝑎1 2𝑑𝑎1 3𝑑 𝑎1 (𝑛 1)𝑑 termThus, the sum of the terms, 𝑆𝑛 , is:𝑆𝑛 𝑎1 (𝑎1 𝑑) (𝑎1 2𝑑) (𝑎1 3𝑑) [𝑎1 (𝑛 1)𝑑]1st2nd3rd4thnthThe terms of an arithmetic sequence can also be written starting from the nthterm and successively subtracting the common difference, 𝑑. Hence,𝑆𝑛 𝑎𝑛 (𝑎𝑛 𝑑) (𝑎𝑛 2𝑑) (𝑎𝑛 3𝑑) [𝑎𝑛 (𝑛 1)𝑑]5
To find the rule for 𝑆𝑛 , add the two equations:𝑆𝑛 𝑎1 𝑆𝑛 𝑎𝑛 (𝑎1 𝑑) (𝑎1 2𝑑) (𝑎1 3𝑑) [𝑎1 (𝑛 1)𝑑] (𝑎𝑛 𝑑) (𝑎𝑛 2𝑑) (𝑎𝑛 3𝑑) [𝑎𝑛 (𝑛 1)𝑑]2𝑆𝑛 (𝑎1 𝑎𝑛 ) (𝑎1 𝑎𝑛 ) (𝑎1 𝑎𝑛 ) (𝑎1 𝑎𝑛 ) (𝑎1 𝑎𝑛 )Notice that all the terms containing d added out. So,2𝑆𝑛 𝑛(𝑎1 𝑎𝑛 )Divide both sides of the equation by two,𝑆𝑛 𝑛(𝑎1 𝑎𝑛 )2Substituting 𝑎𝑛 𝑎1 (𝑛 1)𝑑 to 𝑎𝑛 , will lead to the following formula:𝑛[𝑎1 𝑎1 (𝑛 1)𝑑]2𝑛[2𝑎1 (𝑛 1)𝑑]𝑆𝑛 2𝑆𝑛 Thus, the sum of the terms of an arithmetic sequence is𝑆𝑛 where:𝑛[2𝑎1 𝑑(𝑛 1)]2𝑆𝑛 is the sum of the first n terms𝑎1 is the first termd is the common difference6
WHAT IS ITIn getting the sum of the terms of an arithmetic sequence. We will be usingany of the following the formula:n a1 a n 2n2) S n 2a1 (n 1)d 2Sn 1)if the first and last term are givenif the last term is not givenExample 1. Find the sum of the first 20 natural numbers.Given:𝑎1 1𝑎𝑛 20𝑛 20𝑆𝑛 ?Solution:Since the last term is given, we used the following formula:𝑛𝑆𝑛 2 ( 𝑎1 𝑎𝑛 )Substituting the given values in the formula:𝑆20 202(1 20)𝑆20 10 ( 21 )𝑆20 210 The sum of the first 20 natural numbers is 210.Example 2. Find the sum of the first 16 terms of the arithmetic sequence:8, 11, 14, 17, 20, Given:𝑎1 8𝑛 16𝑑 3𝑆16 ?Solution:The last term is not given, so we use the formula𝑛𝑆𝑛 2 [ 2𝑎1 ( 𝑛 1 ) 𝑑 ]7
By substituting the given values in the formula:𝑆16 162[ 2 ( 8 ) ( 16 1 ) 3 ] 8 [ 16 ( 15) 3 ] 8 ( 16 45 ) 8 ( 61 )𝑆16 488 The sum of the first 16 terms of the series is 488.Example 3. If the first n terms of the sequence: 9, 12, 15, 18, are added,how many terms give a sum of 126?Given:𝑎1 9𝑆𝑛 126𝑑 3𝑛 ?Solution:The last term is not given so we use the following formula𝑛𝑆𝑛 [ 2𝑎1 (𝑛 1)𝑑 ]2Substituting the given:𝑛[2(9) (𝑛 1)3]2𝑛126 [ 18 (3𝑛 3)]2126 252 𝑛 [18 3𝑛 3]252 𝑛 [3𝑛 15]252 3𝑛2 15𝑛0 3𝑛2 15𝑛 252 330 𝑛2 5𝑛 84by factoring(𝑛 12)(𝑛 7) 0(𝑛 12) 0(𝑛 7) 0𝑛 12𝑛 7Since the domain of a sequence is the set of positive integers, we reject𝑛 12. Hence, we only accept 𝑛 7. The number of terms that will add up to 126 is 7.8
Example 4. Find the sum of the integers between 1 and 70, and are divisibleby 3.Given: 𝑎1 3𝑎𝑛 69𝑑 3𝑛 ?𝑆𝑛 ?Solution:a) To solve for 𝑛, use the formula:𝑎𝑛 𝑎1 (𝑛 1)𝑑Substitute the given values:69 3 (𝑛 1)369 3 3𝑛 369 3𝑛𝑛 23b) Since we already solved 𝑛, we can now solve for 𝑆𝑛 .𝑛𝑆𝑛 [ 2𝑎1 (𝑛 1)𝑑 ]223[ 2 ( 3) ( 23 1 ) 3 ]𝑆23 223[ 6 ( 22) 3 ]𝑆23 223𝑆23 ( 6 66 )223𝑆23 ( 72 )2𝑆23 828 The sum of the integers from 1 to 70 that are divisible by 3 is 828.9
Example 5. The sum of the first 15 terms of an arithmetic sequence is 765.If the first term is 23, what is the common difference?Given:𝑎1 23𝑛 15𝑆15 765𝑑 ?Solution:𝑆𝑛 𝑛[ 2𝑎1 (𝑛 1)𝑑 ]2𝑆15 15[2 (23) (15 1)𝑑 ]2765 15[46 ( 14) 𝑑 ]21530 15 (46 14𝑑 )1530 690 210𝑑210𝑑 1530 690210𝑑 840𝑑 4 The common difference is 4.WHAT’S MOREAfter knowing all the needed concept in finding the sum of an arithmeticsequence. You are now ready to answer the following exercises:A. Find the indicated partial sum of each arithmetic stfirst9 terms of 5 8 11 30 terms of 1 3 5 14 terms of 6 9 12 25 terms of 5 8 11 15 terms of 12 ( 6) 0 10
B. Solve for the value of 𝑛.1)2)3)4)𝑆𝑛𝑆𝑛𝑆𝑛𝑆𝑛 80, 50, 15, 180,𝑎1𝑎1𝑎1𝑎1 10,4,12,5,𝑎𝑛 26,𝑎𝑛 16,𝑑 3,𝑑 5,𝑛𝑛𝑛𝑛 ? ? ? ?C) Answer what is asked.1) Find the sum of the first 13 terms of the sequence: 3, 1, 1, 3, 2) Find the sum of the first 15 terms of the arithmetic sequence:10, 15, 20, 25, ?3) Find the sum of the first 11 terms of the arithmetic sequence: 4, 3, 10, 17, ?4) Find the sum of the first 19 terms of the arithmetic sequence:9, 14, 19, 24, ?5) Find the sum of the integers from 8 and 35.6) Find the sum of all even integers from 10 to 70.7) Find the sum of all odd integers from 1 to 50.8) Find the sum of the integers from 20 to 130 and are divisible by 5.9) If the sum of the first 8 terms of an arithmetic sequence is 172 andits common difference is 3, what is the first term?10) If the sum of the first 9 terms of an arithmetic sequence is 216 andits first term is 4, what is the common difference?WHAT I HAVE LEARNEDTo find the sum of the terms of an arithmetic sequence, you can use thefollowing formulae:A. If the first and last terms are given:𝑆𝑛 where:𝑛(𝑎 𝑎𝑛 )2 1𝑆𝑛 is the sum of the first n terms𝑎1 is the first term𝑎𝑛 is the last term11
B. If the last term is not given:𝑆𝑛 where:𝑛[2𝑎1 𝑑(𝑛 1)]2𝑆𝑛 is the sum of the first n terms𝑎1 is the first term𝑑 is the common differenceWHAT I CAN DORead and understand the problems and give what is asked.1.Suppose a cinema has 42 rows of seats and there are 20 seats in the firstrow. Each row after the first row has two more seats that the row that itprecedes. How many seats are in the cinema?2.A 25-layer of logs is being piled to be used on a construction. Theuppermost layer is composed of 25 logs, the second upper layer contains27 logs, and the third upper layer contains 29 logs, and so on. If thepattern continues up to the lowest layer, what is the total number of logspiled for construction?ASSESSMENTRead and analyze each item carefully. Write the letter of the correct answerin a separate paper.1.Which of the following is a formula for arithmetic series?1𝑛1A) 𝑆𝑛 2 (𝑎1 𝑎𝑛 ) B) 𝑆𝑛 2 (𝑎1 𝑎𝑛 ) C) 𝑆𝑛 2 (𝑎1 𝑑)𝑑𝑎𝑛 )12𝑛D)𝑆𝑛 2 (𝑎1
2.Find the sum of the first 12 terms of the arithmetic sequence 4, 11, 18, A) 6103.B) 54C) 34D) 44B) 20C) 15D) 20B) 1720C) 2200D) 6320B) 860C) 435D) 430The first term of an arithmetic sequence is 8, the last term is 56 and thesum is 416. Find the number of terms.A) 139.D) 270Find the sum of the first 15th terms of the arithmetic sequence whosenth term is 𝑎𝑛 5 3𝑛.A) 8708.C) 287Find the sum of the first 40 terms of the arithmetic sequence if the firstterm is 16 and the tenth term is 70.A) 53207.B) 287The second term of an arithmetic sequence is 16 and the eighth term is8. What is the sum of the first 10 terms?A) 156.D) 410The sum of the first 12 terms of an arithmetic sequence is 606. What isthe first term if the last term is 67 and the common difference is 3.A) 645.C) 510Find the sum of the first 15 terms of the arithmetic sequence 17, 12, 7,2, A) 2704.B) 530B) 12C) 11D) 10If the first n terms of the arithmetic sequence 24, 20, 16,. are added,how many of these terms will be added to get a sum of 60?A) 35B) 30C) 25D) 1510. A yaya receives a starting annual salary of Php 60,000.00 with a yearlyincrease of Php 3,600.00. What is her total income for 5 years?A) Php 672,000.00C) Php 276,000.00B) Php 552,000.00D) Php 336,000.0013
11. Jane was saving for a pair of shoes. From her weekly allowance, she wasable to save Php 5.00 on the first week, Php 9.00 on the second,Php 13.00 on the third week, and so on. If she continued saving in thispattern and made 43 deposits, how much did Jane save?A) Php 3,822.00C) Php 7,644.00B) Php 3,827.00D) Php 6,574.0012. Mary gets a starting monthly salary of Php 8,000.00 and an increase ofPhp 800.00 annually. How much income did she receive for the first fouryears?A) Php 441,600.00C) Php 40,000.00B) Php 388,800.00D) Php 36,800.0013. Mirasol saved Php 8.00 on the first day of January, Php 11.00 on thesecond day, Php 14.00 on the third day, and so on, up to the last day ofthe month. How much did Mirasol save at the end of January?A) Php 4,282.00C) Php 1,643.00B) Php 4,290.00D) Php 1,590.0014. Mrs. De la Cruz started her business with an income of Php 250,000.00for the first year and an increase of Php 6,000.00 yearly. How much isthe total income of Mrs. De la Cruz for 6 years since she started herbusiness?A) Php 530,000.00C) Php 1,590,000.00B) Php 3,180,000.00D) Php 1,608,000.0015. A hall has 35 rows. Each successive row contains two additional seats. Ifthe first row has 20 seats, how many seats are in the hall?A) 1,080B) 1,100C) 1,92514D) 1,890
ADDITIONAL ACTIVITYLet us sing the song titled “Twelve Days of Christmas.” Afterwards, answerthe question that follows.Verse 1:Five golden ringsOn the first day of Christmas my true lovesent to meFour calling birdsA partridge in a pear treeVerse 2:On the second day of Christmas my truelove sent to meTwo turtle doves, and a partridge in a peartreeVerse 3:On the third day of Christmas my true lovesent to meThree French hensTwo turtle doves, and a partridge in a peartreeVerse 4:On the fourth day of Christmas my truelove sent to meFour calling birdsThree French hensTwo turtle doves, and a partridge in a peartreeThree French hensTwo turtle doves, and a partridge in a peartreeVerse 7:On the seventh day of Christmas my truelove sent to meSeven swans a - swimmingSix geese a - layingFive golden ringsFour calling birdsThree French hensTwo turtle doves, and a partridge in a peartreeVerse 8:On the eighth day of Christmas my truelove sent to meEight maids a-milkingSeven swans a - swimmingSix geese a - layingVerse 5:Five golden ringsOn the fifth day of Christmas my true lovesent to meFour calling birdsFive golden ringsFour calling birdsThree French hensTwo turtle doves, and a partridge in a peartreeThree French hensTwo turtle doves, and a partridge in a peartreeVerse 6:On the six day of Christmas my true lovesent to meSix geese a – laying15
Verse 9:Verse 11:On the ninth day of Christmas my true lovesent to meOn the 11th day of Christmas my true lovesent to meNine ladies dancing11 pipers pipingEight maids a-milking10 lords a-leapingSeven swans a - swimmingNine ladies dancingSix geese a - layingEight maids a-milkingFive golden ringsSeven swans a - swimmingFour calling birdsSix geese a - laying Five golden ringsThree French hensFour calling birdsTwo turtle doves, and a partridge in a peartreeThree French hensVerse 10:Two turtle doves, and a partridge in a peartreeVerse 12:On the tenth day of Christmas my true lovesent to me10 lords a-leapingOn the 12th day of Christmas my truelove sent to me12 drummers drummingNine ladies dancing11 pipers pipingEight maids a-milking10 lords a-leapingSeven swans a - swimmingSix geese a - layingNine ladies dancingEight maids a-milkingFive golden ringsSeven swans a - swimmingFour calling birdsSix geese a - layingThree French hensFive golden ringsTwo turtle doves, and a partridge in a peartreeFour calling birdsThree French hensTwo turtle doves, and a partridge in a peartreeSummarizing, we have the following:12 drummers drumming6 geese – a – laying11 pipers piping5 golden rings10 lords-a-leaping4 calling birds9 ladies dancing3 French hens8 maids- a – milking2 turtles doves, and7 swans-a-swimmingA partridge in a pear tree.Question:How many gifts are given after the 12th day of Christmas?16
What I Know1. D2. A3. A4. D5. A6. D7. B8. D9. B10. B11. C12. B13. C14. C15. A17What’s MoreA.1) 1532) 9003) 3574) 1 0255) 450B.1) 102) 53) 104) 8C)1. 1172. 6753. 3414. 1 0265. 6026. 1 2407. 6258. 1 7259. 1110. 5What I Can Do1. 2562 seats2. 1225 logs.Assessment1) B2) C3) D4) C5) B6) A7) C8) A9) D10) D11) B12) A13) C14) C15) DAdditional ActivityCallanta, Melvin M., et al., Mathematics Learner’s Module.Pasig City, 2015Nivera, Gladys C. and Lapinid, Minie Rose C.,Grade 10 Mathematics: Patterns andPracticalities. Makati City, Don Bosco Press, 2015. 78 giftsREFERENCES:ANSWER KEY
REFERENCES:Callanta, Melvin M., et al., Mathematics Learner’s Module.Pasig City, 2015Nivera, Gladys C. and Lapinid, Minie Rose C.,Grade 10 Mathematics: Patterns andPracticalities. Makati City, Don Bosco Press, 2015.18
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1. It is the sum of the terms of a sequence. A) mean B) sequence C) nth term D) series 2. Find the sum of the first ten terms of the arithmetic sequence 4, 10, 16, A) 310 B) 430 C) 410 D) 390 3. Find the sum of the first 25 terms of the arithmetic sequence
Texts of Wow Rosh Hashana II 5780 - Congregation Shearith Israel, Atlanta Georgia Wow ׳ג ׳א:׳א תישארב (א) ׃ץרֶָֽאָּהָּ תאֵֵ֥וְּ םִימִַׁ֖שַָּה תאֵֵ֥ םיקִִ֑לֹאֱ ארָָּ֣ Îָּ תישִִׁ֖ארֵ Îְּ(ב) חַורְָּ֣ו ם
Teacher’s Book B LEVEL - English in school 6 Contents Prologue 8 Test paper answers 10 Practice Test 1 11 Module 1 11 Module 2 12 Module 3 15 Practice Test 2 16 Module 1 16 Module 2 17 Module 3 20 Practice Test 3 21 Module 1 21 Module 2 22 Module 3 25 Practice Test 4 26 Module 1 26 Module 2 27 Module 3 30 Practice Test 5 31 Module 1 31 Module .
3. Credit by Quarter (as correlated to district curriculum map): Quarter 1 Module– September and October Quarter 2 Module– November and December Quarter 3 Module– January and February Quarter 4 Module– March and April 4
WinDbg Commands . 0:000 k . Module!FunctionD Module!FunctionC 130 Module!FunctionB 220 Module!FunctionA 110 . User Stack for TID 102. Module!FunctionA Module!FunctionB Module!FunctionC Saves return address Module!FunctionA 110 Saves return address Module!FunctionB 220 Module!FunctionD Saves return address Module!FunctionC 130 Resumes from address
XBEE PRO S2C Wire XBEE Base Board (AADD) XBEE PRO S2C U.FL XBEE Pro S1 Wire RF & TRANSRECEIVER MODULE XBEE MODULE 2. SIM800A/800 Module SIM800C Module SIM868 Module SIM808 Module SIM7600EI MODULE SIM7600CE-L Module SIM7600I Module SIM800L With ESP32 Wrover B M590 MODULE GSM Card SIM800A LM2576
Capacitors 5 – 6 Fault Finding & Testing Diodes,Varistors, EMC capacitors & Recifiers 7 – 10 Fault Finding & Testing Rotors 11 – 12 Fault Finding & Testing Stators 13 – 14 Fault Finding & Testing DC Welders 15 – 20 Fault Finding & Testing 3 Phase Alternators 21 – 26 Fault Finding & Testing
RESUME OF EFFORTS TO SAID LEASE SECTION 19: The Northeast Quarter of the Southwest Quarter of the Southeast Quarter; and The Northwest Quarter of the Southeast Quarter of the Southeast Quarter. Checked probate records in Nevada County. Checked old leases, royalty payment records, and peo
Approaches to Language Teaching: Foundations Module 1: Contextualizing Language Module 2: Building Language Awareness Module 3: Integrating Skills Module 4: Pairwork / Groupwork Module 5: Learner Feedback Approaches to Language Teaching: Extension Module 6: Managing Large Classes Module 7: Learning Strategies Module 8: Authentic Materials Module
