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- 1.
- 2. WHAT SHOULD BETHE THIRD FIGURE?
- 3. WHAT IS THEUNKNOWN FIGURE?
- 4. WHAT IS THEMISSING FIGURE?
- 5. WHAT IS ASEQUENCE? It is a set of numbers which are written in some particular order. For example, take the numbers 1, 3, 5, 7, 9, . . .
- 6. Here is anothersequence: 1, 4, 9, 16, 25, . . . And this sequence, 1, −1, 1, −1, 1, −1, . . .
- 7. The numbers 1,3, 5, 7, 9 form a finite sequence containing just five numbers. The numbers 1, 4, 9, 16, 25 also form a finite sequence.
- 8. 1, 2, 3,4, 5, 6, . . ., n
- 9. A sequence isfinite if it has a limited number of terms and infinite if it does not.
- 10. INFINITE OR FINITE? 1.2, 7, 12, 17, 22, 27 a, c, e, g,… 2. 2, 4, 8, 16, 32,… -1, 4, 3, 7, 10, 17,…71
- 11. ARITHMETIC SEQUENCE Consider thesecommon sequences 1, 3, 5, 7, . . . 0, 10, 20, 30, 40, . . . 8, 5, 2, −1, −4, . . .
- 12. BASED ON THEEXAMPLES ABOVE Can you define arithmetic sequence?
- 13. An arithmetic sequenceis a sequence where every term after the first term is obtained by adding a constant called the common difference.
- 14. We can usealgebraic notation to represent an arithmetic sequence. We shall let a1 stand for the first term of the sequence, and let d stand for the common difference between successive terms.
- 15. For example, thesequence 1, 3, 5, 7, 9,…could be written as a1, a1 + d, a1 + 2d, a1 + 3d, a1 + 4d, a1 + (n-1)d, … where a1 = 1 is the first term, and d = 2 is the common difference.
- 16. If we wantedto write down the nth term of an arithmetic sequence, we would have an=a1+(n−1)d We use the nth term to find the value of a specific term of an arithmetic sequence based on its position.
- 17. FOR EXAMPLE Find the25th term of the sequence 3/5, 19/15, 29/15,… d = a2 - a1 d = - d = d = or an=a1+(n−1)d a25=+(25−1) a25=+(24) a25= +16 = =
- 18. From the nth termof an arithmetic sequence, we can derive the General/ Explicit term/formula of arithmetic sequence an= dn + a1−d
- 19. EXERCISES Give the commondifference and the missing terms of the arithmetic sequences below. 1) -53, ___, -37, -29, ____, -13, ____, 3 2) , ___, , ___, , ,… 3) , ___, , , ___, ___.
- 20. EXERCISES Find the twelfthterm of a sequence where the seventh term is 10 and the common difference is −2. Give the formula for the general/explicit term.
- 21. SOLUTION First find thefirst term, a1, use the values a7=10, n=7 , and d=−2. Substitute in the nth term formula and simplify. an=a1+(n−1)d 10=a1+(7−1)(−2) 10=a1+(6)(−2) 10=a1−12 a1=22
- 22. Find the twelfthterm, a12, using the formula with a1=22, n=12, and d=−2. an=a1+(n−1)d a12=22+(12−1)(−2) a12=22+(11)(−2) a12=0
- 23. HOW ABOUT GENERAL/EXPLICITTERM OF THE SEQUENCE To find the general term/explicit term, substitute the values into the formula, a1 = 22 and d= -2 an = dn + a1−d an = -2n + 22− (-2)
- 24. Find the firstterm and the common difference of a sequence where the fifth term is 19 and the eleventh term is 37. Give the formula for general/explicit term.
- 25. Since we knowtwo terms, we can make a system of equations using the formula for the general term. an = a1+(n−1)d a5 = a1+(5−1)d 19 = a1+4d a11 = a1+(11−1)d 37 = a1+10d _____________________________________________________________ -18 = 0 -6d 3 = d
- 26. Substituting d=3 ineither of the two equations above 19 = a1+4d 37 = a1+10d 19 = a1 +4 (3) 37 = a1+10(3) 19 = a1 + 12 37 = a1+30 7 = a1 7 = a1
- 27. How about thegeneral term Use the values a1 = 7 and d = 3 an = dn + a1−d an = 3n + 7 - 3 an = 3n + 4
- 28. Find the 19th termand the common difference of a sequence where the fifth term is 19 and the eleventh term is 37.
- 29. SOLUTION We can directlysolve the common difference using the formula an = ak+(n−k)d a5 = 19 and a11 = 37 a11 = a5+(n−5)d 37 = 19+(11−5)d 37-19 = 6d 18 = 6d 3 = d
- 30. Find the 19th termusing the formula an = ak+(n−k)d a5 = 19, and d = 3 a19 = a5+(n−5)3 a19 = 19+(19−5)3 a19 = 19 + 42 a19 = 61
- 31. Find the 19th termusing the formula an = ak+(n−k)d a11 = 37, and d = 3 a19 = a11+(n−11)3 a19 = 37+(19−11)3 a19 = 37 + 24 a19 = 61
- 32. Find the thirdterm and the common difference of a sequence where the fourth term is 17 and the thirteenth term is 53. Give the explicit formula of the sequence
- 33. SOLUTION an = a1+(n−1)d a4= a1+(4−1)d 17 = a1+3d a13 = a1+(13−1)d 53 = a1+12d __________________________________________________________________________________ -36 = 0 - 9d 4 = d
- 34. Substituting d=4 ineither of the two equations above 17 = a1+3d 53 = a1+12d 17 = a1 +3(4) 53 = a1+12(4) 17 = a1 + 12 53 = a1+48 5 = a1 5 = a1
- 35. Explicit formula/General Term.Use the values a1 = 5 and d = 4 an = dn + a1−d an = 4n + 5 −4 an = 4n+1
- 36. How many termsare in an arithmetic sequence whose first term is -3 , common difference is 2, and the last term is 23?
- 37. SOLUTION an = a1+(n−1)d 23=-3+(n−1)2 23 + 3= 2n -2 26+2 = 2n 28 = 2n 14 = n
- 38. What must bethe value of k so that 5k – 3, k + 2, 3k – 11 will form an arithmetic sequence?
- 39. Since the givensequence is arithmetic, a2-a1 = a3 -a2 k + 2 – (5k -3) = 3k-11 – (k + 2) k + 2 -5k + 3 = 3k -11 -k -2 -4k+5 = 2k -13 -4k -2k = -13 – 5 -6k = -18 k = 3
- 40. if k =3 then, 5k - 3, k + 2, 3k-11 12, 5, -2
- 41. RECURSIVE FORMULA OFAN ARITHMETIC SEQUENCE
- 42. RECURSIVE FORMULA A recursiveformula for an arithmetic sequence with common difference d is given by an=an-1+d, where n≥2
- 43. RECURSIVE FORMULA Consider thesequence 3, 5, 7,... an=an-1+d Value of n an-1 +d an-1+d an=an-1+d n=2 = a2-1 + 2 a1 + 2 a2=3+2 = 5 n=3 = a3-1 + 2 a2 + 2 a3=5+2 = 7 n=4 = a4-1 + 2 a3 + 2 a4=7+2 = 9 n=5 = a5-1+ 2 a4 + 2 a5=9+2 = 11 This formula gives us the same sequence as described by 3,5,7,... So the recursive formula is an=an-1+ 2
- 44. The main differencebetween recursive and explicit is that a recursive formula gives the value of a specific term based on the previous term while an explicit formula gives the value of a specific term based on the position.
- 45. EXERCISES Given two termsof an arithmetic sequence, find the recursive formula. a18=3362 and a38=7362
- 46. SOLUTION an = a1+(n−1)d a18= a1+(18−1)d 3362 = a1+17d a38 = a1+(38−1)d 7362 = a1+37d _______________________________________________________________________________________________________________________________________________________________ -4000 = 0 - 20d 200 = d So the recursive formula is : an = an-1+200
- 47. EXERCISES Given two termsof an arithmetic sequence, find the recursive formula. a2=4/15 and a4= -2/5
- 48. SOLUTION an = a1+(n−1)d a2= a1+(2−1) d 4/15 = a1+ d a4 = a1+(4−1)d -2/5 = a1+3d _______________________________________________________________________________________________________________________________________________________________ 10/15 = 0 + -2d -1/3 = d So the recursive formula is : an = an-1-1/3
- 49. ARITHMETIC MEANS The termsbetween any two nonconsecutive terms of an arithmetic sequence.
- 50. Insert 4 arithmeticmeans between 10 and -15. Solution. Since we are tasked to insert 4 terms, then there will be 6 terms in all. an = a1+(n−1)d 10, a2, a3, a4, a5, -15 a6 = 10+(6−1)d -15 = 10+5d -15 -10 = 5d -25 = 5d -5 = d 10, 5, 0. -5, -10, -15
- 51. Find the valueof x if the arithmetic mean of 3 and 3x – 5 is 8.
- 52. SOLUTION Since the arithmeticmean is 8, we have 3, 8, 3x + 5 a2-a1 = a3 -a2 8-3 = 3x+5 – 8 5= 3x -3 5 + 3 = 3x 8 = 3x 8/3 = x
- 53. if x =8/3 then, 3, 8, 3x + 5 3, 8, 3(8/3) + 5 3, 8, 13
- 54. EXERCISES The arithmetic meanbetween two terms in an arithmetic sequence is 39. if one of these terms is 32, find the other term.
- 55. SOLUTION 32, 39,____. Or____, 39, 32 d=7 d=-7 an = a1+(n−1)d an = a1+(n−1)d a3 = 32+(3−1)7 a3 = a1 +(3 −1)7 a3 = 32+ 14 32 = a1 − 14 a3 = 46 a1 = 46
- 56. Find the valueof a when the arithmetic mean of a+7 and a+3 is 3a+9.
- 57. SOLUTION Since the arithmeticmean is 3a+9, we have a+7, 3a+9, a+3 a2-a1 = a3 -a2 3a+9 – (a+7) = a+3 – (3a+9) 3a+9 – a-7 = a+3 – 3a-9 2a+2 = -2a-6 4a = -8 a = -2
- 58. If a =-2, then the sequence is a+7, 3a+9, a+3 5, 3, 1
- 59. THE SUM SNOF THE FIRST N TERMS OF ANY ARITHMETIC SEQUENCE The sum Sn of the first n terms of any arithmetic sequence is written as Sn = a1 + a2 +… + an
- 60. FORMULA TO FINDTHE SUM OF AN ARITHMETIC SEQUENCE we can write Sn = a1 + a2 +… + an as Sn = a1 + (a1 + d) + (a1 +2d) +… +an Sn = an + (an - d) + (an -2d) +… +a1 _______________________________________________ 2Sn = (a1 + an) + (a1 + an) + (a1 + an) +… + (a1 + an)
- 61. 2Sn = (a1+ an) + (a1 + an) + (a1 + an) +… + (a1 + an) Since there are n terms of the form a1 + an, then 2Sn = n(a1 + an) By dividing both sides by 2, we have Sn =
- 62. Since we alsoknow that an = a1 + (n-1)d, then by substitution we have, Sn = Sn = ) simplify, Sn =
- 63. EXERCISES The third termof an arithmetic sequence is -12 and the seventh term is 6. What is the sum of the first 10 terms? Find the explicit and recursive formula.
- 64. SOLUTION an = a1+(n−1)d a3= a1+(3−1)d -12 = a1+2d a7 = a1+(7−1)d 6 = a1+6d ______________ -18 = 0 - 4d = d or = d
- 65. Substituting d= ineither of the two equations above a3 = a1+(3−1) a7 = a1+(7−1)d -12 = a1 +(2) 6 = a1+(6) -12 = a1 + 9 6 = a1+27 -12-9 = a1 6-27= a1 -21 = a1 -21= a1
- 66. = -21 d= Sn = S10 = S10 = 5 S10 = 5 S10 = 5 S10 = 5 S10 =
- 67. Explicit formula n + n- Recursive Formula an=an-1+d an=an-1+
- 68. SUMMATION NOTATION The sumof 4n as n goes from 1 to 6
- 69.
- 71. SOLUTION a1 = 11and d = 4
- 72. Sn = S25 = S25= S25 = S25 = S25 = 25 ( 59) S25 = 1475
- 73.
- 74. GROUP ACTIVITY 1. Solvethe following problems. a)A construction company will be penalized each day of delay in construction for bridge. The penalty will be 4000 for the first day and will increase by 1000 for each following day. Based on its budget, the company can afford to pay a maximum of 165000 toward penalty. Find the maximum number of days by which the completion of work can be delayed. b)Find the sum
- 75. 2. A doctorprescribed 15 pills for his patient to be taken in the first week.Given that the patient should decrease the dosage by 3 pills every week,find the week in which he will stop taking the medicine completely. Ramy saves $1 on the first day, $2 on the second day, $3 on the third day,and so on, saving an extra £1 each day.On which day will he have saved over $100 in total?
- 76. 3. A companywants to distribute ₱ 14 500 among the top 5 sales representatives as a bonus. The bonus for the last-place representative is ₱1 300, and the difference in bonus is constant among the representatives. Find the bonus of the representative in the first place. Find the sum of the first 12 terms of the arithmetic sequence whose general term is an = 3n + 5
- 77. 4. Find thevalue of x when the arithmetic mean of x + 2 and 4x + 5 is 3x + 2. The sum of the first 10 terms of an arithmetic sequence is 530. What is the first term if the last term is 80? What is the common difference?
- 78. ASSIGNMENT How many integersbetween 60 and 600 are divisible by 2 or by 3? Apply the concept of arithmetic sequence.
- 79. MATEO16:26 Sapagkat ano ngaba ang mapapala ng isang tao makamtan man nya ang buong daigdig, ngunit mapapahamak naman ang kanyang sarili?
- 80.
- 84. JUAN 14:6 Ako angdaan, ang katotohanan, at ang buhay. Walang makakarating sa Ama kung hindi sa pamamagitan.
- 85.
Editor's Notes
- #2 This time we are going to apply the concept of sequence in abstract reasoning.
- #5 Ok, let us first define sequence. As you can see, We have a sequence of odd numbers. We start with the number 1, which is an odd number, and then each successive number is obtained by adding 2 to give the next odd number.
- #6 This is the sequence of square numbers. And this sequence is a sequence of numbers alternating between 1 and −1 In each case, the dots written at the end indicate that we must consider the sequence as an infinite sequence. And so it goes on forever. Do you believe in forever? If you don’t, in infinte sequence, there is always forever.
- #7 On the other,
- #8 In this sequence, the dots indicate that we have not explicitly written down all the numbers. The n after the dots tells us that this is a finite sequence, and the last number is n.
- #10 Ok let us try these two numbers
- #11 Can you see the pattern? how do these sequences formed. It is easy to see how these sequences are formed. They start with a particular first term, and then to get successive terms we just add a fixed value to the previous term. In the first sequence we add 2 to get the next term, and in the second sequence we add 10 and in the 3rd sequence we add -3.
- #16 This is the nth term of arithmetic sequence. It is also called general term or general formula of arithmetic sequence. Other reference called it explicit formula.
- #18 This is the nth term of arithmetic sequence. It is also called general term or general formula of arithmetic sequence. Other reference called it explicit formula.
- #19 Answer 1) -45 -21 -5 d= 8 2) 8/9 14/9 d = 3/9 or 1/3 3) 37/12 13/12 5/12 d= -8/12 or -2/3
- #49 How to insert arithmetic means?
- #55 Our sequence is either 32, 39,____. Or ____, 39, 32 Where the common difference in the first sequence is 7 and the -7 in the second sequence.
- #59 How about if there are 100 terms in an arithmetic sequence, are we going to add them one by one? Adding all terms of an arithmetic sequence can be tedious, so we have to develop formula to find the sum of arithmetic sequence.