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Arithmetic Sequence and Arithmetic Series

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Arithmetic Sequence and Arithmetic Series

  1. 1. Arithmetic Sequence and Arithmetic Series 1
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  3. 3. At the end of the lesson, the learners are expected to: 1) Generate and describe patterns using symbols and mathematical expressions; 2) Find the next few terms and the nth term of the given sequence; 3) Define and describe an arithmetic sequence; 4) Enumerate the next few terms and the nth term of an arithmetic sequence; 5) Insert means between two given terms of an arithmetic sequence; 6) Find the sum of the first n terms of an arithmetic sequence; and 7) Solve problems involving arithmetic sequence. 3
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  7. 7. 1. O T T F F S S E N _? T E F S 2. R O Y B G I _? T U V W 3. J F M A M J J A S O N _? A B C D 4. T Q P H H O N _? D U H I Direction: Click the letter of your answer. 7
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  10. 10. 1) 80, 40, 20, 10, 5, __? 2) 1, 8, 27, 64, 125, 216, __? 3) 1, 4, 9, 16, 25, 36, 49, __? 343 512 729 810 56 64 72 81 0 1 2 1 5 2 13
  11. 11. 4) 1, 2, 3, 5, 8, 13, 21, __? 5) -2, -5, -8, -11, -14, __? 6) 7, 11, 15, 19, 23, 27, __? -17 -19 -21 -23 29 30 31 32 31 32 33 34 14
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  25. 25. An ordered list of numbers, such as 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, … is called a sequence. A number sequence is a list of numbers having a first number, a second number, a third number, and so on, called the terms of the sequence. 32
  26. 26. FINITE AND INFINITE SEQUENCES • 10, 15, 20, 25, 30 • There are only a finite number of terms. • They have a last term. • 11, 22, 33, 44, 55, … • There are infinite number of terms. • They do not have a last term. 33
  27. 27. Sequences are usually given by stating their general or nth term. NUMBER SEQUENCE Example: Consider the sequence given 𝒂 𝒏 = 𝟑𝒏 + 𝟐 The first five terms of the sequence are 𝒂 𝟏 = 𝟑 𝟏 + 𝟐 = 𝟑 + 𝟐 = 𝟓, 𝒂 𝟐 = 𝟑 𝟐 + 𝟐 = 𝟔 + 𝟐 = 𝟖, 𝒂 𝟑 = 𝟑 𝟑 + 𝟐 = 𝟗 + 𝟐 = 𝟏𝟏, 𝒂 𝟒 = 𝟑 𝟒 + 𝟐 = 𝟏𝟐 + 𝟐 = 𝟏𝟒, 𝒂𝒏𝒅 𝒂 𝟓 = 𝟑 𝟓 + 𝟐 = 𝟏𝟓 + 𝟐 = 𝟏𝟕 34
  28. 28. a. 𝒂 𝒏 = 𝟐𝒏 𝟐 for 𝟏 ≤ 𝒏 ≤ 𝟓 b. 𝒂 𝒏 = 𝟐𝒏 − 𝟏 for 𝟏 ≤ 𝒏 ≤ 𝟓 c. 𝒂 𝒏 = (−𝟏) 𝒏 (𝒏 − 𝟑) 𝟐 for 𝟏 ≤ 𝒏 ≤ 𝟒 List all the indicated terms of each finite sequence. NUMBER SEQUENCE 35
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  30. 30. QUESTION: What do you observe in the following pictures 38
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  32. 32. Joey’s school offered him a scholarship grant of ₱ 3,000.00 when he was in Grade 7 and increased the amount by ₱ 500 each year till Grade 10. The amounts of money (in ₱) Joey received in Grade 7, 8, 9, and 10 were respectively: 3000, 3500, 4000, and 4500 Each of the numbers in the list is called a term. Note: We find that the succeeding terms are obtained by adding a fixed number. 40
  33. 33. In a savings scheme, the amount triples after every 7 years. The maturity amount (in ₱) of an investment of ₱ 6,000.00 after 7, 14, 21 and 28 years will be, respectively: 18000, 54000, 162000, 486000 Note: We find that the succeeding terms are obtained by multiplying with a fixed number. 41
  34. 34. NUMBER PATTERNS The number of unit squares in a square with sides 1, 2, 3, 4, ... units are respectively 1, 4, 9, 16, .... Note: We can observe that 1=12, 4=22, 9=32, 16=42, ... Thus, the succeeding terms are squares of consecutive numbers. 42
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  36. 36. ARITHMETIC SEQUENCE Consider the following lists of numbers : 2, 4, 6, 8, 10, .... 15, 12, 9, 6, 3, .... -5, -4,-3, -2, -1.... 6, 6, 6, 6, 6, 6, .... each term is obtained by adding 2 to the previous term each term is obtained by adding -3 to the previous term each term is obtained by adding 1 to the previous term each term is obtained by adding 0 to the previous term 44
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  38. 38. 1) 23, 38, 53, __ , 83, 98 2) 45, 37, __ , 21, 13, 5 3) -13, -6, __ , 8, 15, 22 27 28 29 30 0 1 -2 -3 63 68 73 78 46
  39. 39. 4) __ , 23, 32, 41, 50, 59 5) -12, -7, -2, 3, 8, ___ 6) 10, __ , 32, 43, 54, 65 10 11 12 13 21 23 25 27 10 12 14 16 47
  40. 40. The constant is called the common difference, denoted with the letter d, referring to the fact that the difference between two successive terms yields the constant value that was added. To find the common difference, subtract the first term from the second term. In mathematics, an arithmetic sequence or arithmetic progression is a sequence of numbers where each term after the first term is obtained by adding the same constant (always the same). 52
  41. 41. Examples: 5, 10, 15, 20, 25, … 3, 7, 11, 15, … 20, 18, 16, 14, … -7, -17, -27, -37, … d=5 d=4 d=-2 d=-10 53
  42. 42. 2, 5, 8, 11, 14 … In this sequence, notice that the common difference is 3. In order to get to the next term in the sequence, we will add 3; so, a recursive formula for this sequence is: 31  nn aa The first term in the sequence is a1 (sometimes just written as a). Example: 54
  43. 43. 2, 5, 8, 11, 14 … +3 +3 +3 +3 Every time you want another term in the said sequence of numbers, you have to add d. This means that the second term was the first term plus d. The third term is the first term plus d plus d (added twice). The fourth term is the first term plus d plus d plus d (added three times). So, you can see to get the nth term we have to take the first term and add (n - 1) times d. 𝑑 = 3  dnaan 1 𝑎 = 2   1715231626 a 55
  44. 44. To find any term of an arithmetic sequence: where: a1 is the first term of the sequence, d is the common difference, n is the number of the term to find. ARITHMETIC SEQUENCE 56
  45. 45. Find the nth term of the arithmetic sequence 5, 8, 11, 14, … We know that 𝒂 𝟏 = 𝟓 𝒅 = 𝟑 Substituting in the formula, we obtain 𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑 𝑎 𝑛 = 5 + 𝑛 − 1 3 𝑎 𝑛 = 5 + 3𝑛 − 3 𝒂 𝒏 = 𝟐 + 𝟑𝒏 57
  46. 46. Find the 18th term of the arithmetic sequence 21, 24, 27, 30, 33, … Note that 𝒂 𝟏 = 𝟓, 𝒅 = 𝟑, 𝒏 = 𝟏𝟖. Using the formula for the general term of an arithmetic sequence, we have 𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑 𝑎 𝑛 = 21 + 18 − 1 3 𝑎 𝑛 = 21 + (17)3 𝒂 𝒏 = 𝟕𝟐 Thus, 72 is the 18th term of the sequence. 58
  47. 47. In the arithmetic sequence 10, 16, 22, 28, 34, … , which term is 124? The problem asks for n when 𝑎 𝑛 = 124. From the given sequence 𝒂 𝟏 = 𝟏𝟎, 𝒅 = 𝟔, 𝒂 𝒏 = 𝟏𝟐𝟒. Substitute these values in the formula to get 𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑 124 = 10 + 𝑛 − 1 6 124 = 10 + 6𝑛 − 6 120 = 6𝑛 𝟐𝟎 = 𝒏 Thus, 124 is the 20th term. 59
  48. 48. Find the first term of the arithmetic sequence with a common difference of 6 and whose 20th term is 965? Since 𝒂 𝟐𝟎 = 𝟗𝟔𝟓, 𝒅 = 𝟔, 𝒏 = 20 , we have 𝑎20 = 𝑎1 + 𝑛 − 1 𝑑 965 = 𝑎1 + 20 − 1 6 965 = 𝑎1 + 19 6 965 = 𝑎1 + 114 𝑎1 = 965 − 114 = 851 Thus, 851 is the first term. 60
  49. 49. Find the 15th term of the arithmetic sequence whose first term is 7 and whose 5th term is 19. Find the common difference, d by substituting 𝑎1 = 7 and, 𝑎5 = 19 . Thus, 𝑎5 = 𝑎1 + 𝑛 − 1 𝑑 19 = 7 + 5 − 1 𝑑 19 = 7 + 4𝑑 12 = 4𝑑 3 = 𝑑 Now use 𝒂 𝟏 = 𝟕, 𝒅 = 𝟑, 𝒏 = 𝟏𝟓 𝒂 𝟏𝟓 = 𝟕 + 𝟏𝟓 − 𝟏 𝟑 = 𝟒𝟗 Thus, the 15th term is 49. 61
  50. 50. Insert two arithmetic means between 8 and 17. Our task here is to find 𝑎2 and 𝑎3 such that 8, 𝑎2, 𝑎3, 17 form an arithmetic sequence. 𝑎1 = 8 𝑎4 = 17 𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑 17 = 8 + 4 − 1 𝑑 17 = 8 + 3𝑑 15 = 3𝑑 5 = 𝑑 We need 𝑎2 and 𝑎3 . Hence, 𝑎2 = 8 + 3 = 𝟏𝟏 and 𝑎3 = 8 + 3 + 3 = 𝟏𝟒 62
  51. 51. A conference hall has 33 rows of seats. The last row contains 80 seats. If each row has two fewer seats than the row behind it. How many seats are there in the first row? We know that 𝒂 𝟑𝟑 = 𝟖𝟎, 𝒅 = 𝟐, 𝒏 = 33 . Using the formula for the general term of an arithmetic sequence, we have 𝑎33 = 𝑎1 + 𝑛 − 1 𝑑 80 = 𝑎1 + 33 − 1 2 80 = 𝑎1 + 32 2 80 = 𝑎1 + 64 𝑎1 = 80 − 64 = 16 Therefore, there are 16 seats in the first row. 63
  52. 52. How many numbers between 7 and 500 are divisible by 5? The common difference, d, is 5. Since we want numbers that are divisible by 5, then we let 𝒂 𝟏 = 𝟏𝟎, 𝒂 𝒏 = 𝟓𝟎𝟎. Substitute these values into the general term. 𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑 500 = 10 + 𝑛 − 1 5 500 = 10 + 5𝑛 − 5 495 = 5𝑛 𝟗𝟗 = 𝒏 There are 99 numbers between 7 and 500 that are divisible by 5. 64
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  54. 54. 1. 1, 2, 3, 4, … and 1+2+3+4+… 2. 2, 4, 6, 8, … and 2+4+6+8+… 3. 5, 10, 15, … and 5+10+15+… 4. 3, 7, 11, 15, … and 3+7+11+15+… 5. 4, 8, 12, 16, … and 4+8+12+16+… Each indicated sum of the terms of an arithmetic sequence is an arithmetic series. 66
  55. 55. Do you know this? The story is told of a grade school teacher In the 1700’s that wanted to keep her class busy while she graded papers so she asked them to add up all of the numbers from 1 to 100. These numbers are an arithmetic sequence with common difference 1. Carl Friedrich Gauss was in the class and had the answer in a minute or two (remember no calculators in those days). This is what he did: 1 + 2 + 3 + 4 + 5 + . . . + 96 + 97 + 98 + 99 + 100 sum is 101 sum is 101 67
  56. 56. The formula for the sum of n terms is:  nn aa n S  1 2 n is the number of terms so 𝒏 𝟐 would be the number of pairs Let’s find the sum of 1 + 3 +5 + . . . + 59 ARITHMETIC SERIES first term last term 68
  57. 57.  nn aa n S  1 2 Let’s find the sum of 1 + 3 +5 + . . . + 59  12 nThe common difference is 2 and the first term is 1, so: Set this equal to 59 to find n. Remember n is the term number. 𝟐𝒏 − 𝟏 = 𝟓𝟗 𝒏 = 𝟑𝟎 So there are 30 terms to sum up.   900591 2 30 30 S first term last term 69
  58. 58. To find the sum of a certain number of terms of an arithmetic sequence: where: Sn is the sum of n terms (nth partial sum), a1 is the first term, an is the nth term. 70
  59. 59. To find the sum of a certain number of terms of an arithmetic sequence: where: Sn is the sum of n terms (nth partial sum), a is the first term, n is the “rank” of the nth term d is the common difference 71
  60. 60. Find the sum of the first ten positive integers. a1 = 1 n = 10 Illustrative Example a10 = 10 𝑆10 = 𝑛 2 (𝑎1 + 𝑎 𝑛) 𝑆10 = 10 2 (1 + 10) 𝑆10 = 5 (11) 𝑆10 = 𝟓𝟓 72
  61. 61. Find the sum of the first 15 terms of the arithmetic sequence if the first term is 11 and the 15th term is 109. a1 = 11 n = 15 Illustrative Example a15 = 109 𝑆15 = 𝑛 2 (𝑎1 + 𝑎 𝑛) 𝑆15 = 15 2 (11 + 109) 𝑆15 = 15 2 (120) 𝑆15 = 𝟗𝟎𝟎 73
  62. 62. Find the sum of all the odd integers from 1 to 99. a1 = 1 d = 2 Illustrative Example Here, a10 = 99 𝑆50 = 𝑛 2 (𝑎1 + 𝑎 𝑛) 𝑆50 = 50 2 (1 + 99) 𝑆50 = 25 (100) 𝑆50 = 𝟐, 𝟓𝟎𝟎 𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑 99 = 1 + 𝑛 − 1 2 99 = 1 + 2𝑛 − 2 100 = 2𝑛 50 = 𝑛 74
  63. 63. In a classroom of 40 students, each student counts off by fours (i.e. 4, 8, 12, 16, …). What is the sum of the students’ numbers? a1 = 4 d =4 Illustrative Example Here, n=40 𝑆40 = 𝑛 2 (𝑎1 + 𝑎 𝑛) 𝑆40 = 40 2 (4 + 160) 𝑆40 = 20 (164) 𝑆40 = 𝟑, 𝟐𝟖𝟎 𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑 𝑎 𝑛 = 4 + 40 − 1 4 𝑎 𝑛 = 4 + 156 𝑎 𝑛 = 160 75
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  66. 66. Practice Exercises Determine if the given sequence is an arithmetic sequence or not. 1. 6, 8, 10, 12, … 2. -11, -10, -9, -8, … 3. 8, 11, 14, 17, … 4. 5, 15, 45, 135, … 5. 6, 11, 17, 21, … Arithmetic Sequence NOT Arithmetic Sequence Arithmetic Sequence Arithmetic Sequence Arithmetic Sequence NOT NOT NOT NOT Click the figure which corresponds to your answer. 78
  67. 67. Practice Exercises Solve the following problems. 6. In the arithmetic sequence , 2, 5, 8, 11, …, what is the 30th term? 7. In the arithmetic sequence 8, 5, 2, -1, …, what is the 15th term? 8. In the arithmetic sequence with 12 as the first term and the common difference is -3, what is the 17th term? 87 -28 88 89 90 -30 -32 -34 -36 -37 -38 -39 Click the figure which corresponds to your answer. 79
  68. 68. Practice Exercises Solve the following problems. 9. In the arithmetic sequence 23, 30, 37, 44, …, what is the 14th term? 10. In the arithmetic sequence 6, 12, 18, …, what is the 29th term? 110 170 112 114 116 172 174 176 Click the figure which corresponds to your answer. 80
  69. 69. Practice Exercises Find the arithmetic mean of the following numbers. 11) 4 and 16 12) 19 and 35 13) 13 and 25 14) -22 and 8 15)102 and 1002 -6 8 10 12 26 27 28 29 6 15 17 2119 550 -7 552 -8 554 -9 556 Click the figure which corresponds to your answer. 81
  70. 70. Practice Exercises Solve the following problems. 16. What is the sum of the first 100 positive odd integers? 17. What is the sum of the first 50 positive even integers? 18. What is the sum of the first 30 positive multiples of 8? 3730 11000 12000 13000 2500 3710 2550 2600 2650 3720 10000 3740 Click the figure which corresponds to your answer. 82
  71. 71. Practice Exercises 19. Aris takes a job, starting with an hourly wage of ₱ 350.00 and is promised a raise of ₱ 5.00 per hour every two months for 5 years. At the end of 5 years, what would be Aris’ hourly wage? 20. Find the sum of all two-digit even natural numbers ₱ 485 ₱ 490 ₱ 495 ₱ 500 2410 2420 2430 2440 Click the figure which corresponds to your answer. 83
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  73. 73. Assessment Determine if the sequence is an arithmetic sequence or not. 1. 9, 11, 13, 15, … 2. 6, 11, 16, 21, … 3. 1, 2, 4, 8, 16, … 4. -4, 2, 8, 14, … 5. 1, 8, 27, 64, … 97
  74. 74. Assessment Solve the following problems. 6. What is the 11th term in the sequence 6, 9, 12, 15,… ? 7. What is the 24th term of the sequence 16, 19, 22, … ? 8. What is the 25th term of the sequence 12, 9, 6, … ? 9. What is the 30th term in the arithmetic sequence with a first term of 15 and a common difference of 5? 10. What is the 10th term of the arithmetic sequence with a first term of 75 and a common difference of -8? 98
  75. 75. Assessment 11. Insert the arithmetic mean of 8 and 28. 12. Insert two arithmetic means between 16 and 31. 13. Insert two arithmetic means between 21 and 33. 14. Insert three arithmetic means between 11 and 35. 15. Insert three arithmetic means between 48 and 84. 99
  76. 76. Assessment Solve the following problems. 16. Joan started a new job with an annual salary of ₱ 150 000 in 2007. If she receives a ₱ 12 000 raise each year, how much will her annual salary be in 2017? 17. A stack of telephone poles has 30 poles in the bottom row. There are 29 poles in the second row, 28 in the next row, and so on. How many poles are there in the 26th row? 18. Josh spent ₱ 150 on August 1, ₱ 170 on August 2. ₱ 190 on August 3, and so on. How much did Josh spend on August 31? 100
  77. 77. Assessment 19. An object is dropped from a jet plane and falls 32 feet during the first second. If during each successive second, it falls 40 feet more than the distance in the preceding second, how far does it fall during the eleventh second? 20. What is the seating capacity of a movie house with 40 rows of seats if there are 25 seats in the first row, 28 seats in the second row, 31 in the third row, and so on? 101
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  79. 79. • Acelajado, Maxima J. (2008). New High School Mathematics II Second Edition. Makati City: Diwa Learning Systems, Inc. • Callanta, Melvin M., et al. (2015). Mathematics – Grade 10 Learner’s Module. Pasig City: Department of Education. • Orines, Fernando B., et al. (2008). Next Century Mathematics (Intermediate Algebra) Second Edition. Quezon City: Phoenix Publishing House, Inc. • Oronce, Orlando A., Mendoza, Marilyn O. (2010). E-Math II. Manila: Rex Book Store, Inc. • http://www.google.com.ph (Some of the pictures used in this presentation were taken from the said site) • http://www.slideshare.com (Some of the examples and exercises of arithmetic sequence and arithmetic series used in this presentation were taken from the said site) • https://www.youtube.com/watch?v=HlZky0FL6ck (The video used in this presentation was taken from the said site) 108
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