Arithmetic Sequence
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This document defines arithmetic sequences and provides examples and explanations of key concepts such as the common difference, finding specific terms, finding sums of terms, and solving word problems involving arithmetic sequences. Key points covered include defining an arithmetic sequence as a sequence where each term is obtained by adding a constant to the preceding term, using the formula an = a1 + (n-1)d to find the nth term, and the formulas Sn = n(a1 + an)/2 and Sn = n(n-1)d/2 to find the sum of the first n terms. Examples of finding missing terms, common differences, and sums are worked through.
Arithmetic Sequence
- 1. ARITHMETIC SEQUENCE JOEY F. VALDRIZ
- 2. Arithmetic Sequence • a sequence of numbers where each term after the first term is obtained by adding the same constant to the preceding term The constant is called a common difference, denoted as 𝒅.
- 3. Examples of Arithmetic Sequence 1. 6, 10, 14, 18, 22, 26, … 𝑑 = 4 2. 20, 17, 14, 11, 8, 5, … 𝑑 = −3 3. 1 4 , 1 2 , 3 4 , 1, 5 4 , … 𝑑 = 1 4 4. 11 3 , 35 12 , 13 6 , 17 12 , 2 3 , … 𝑑 = − 3 4
- 4. 2, 4, 6, 8, 10, .... 15, 12, 9, 6, 3, .... -5, -4,-3, -2, -1.... 6, 6, 6, 6, 6, 6, .... 𝒅 =? 𝒅 =? 𝒅 =? 𝒅 =? Find the common difference for each arithmetic sequence.
- 5. 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 Identify the missing term in the given arithmetic sequence.
- 6. 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 Identify the missing term in the given arithmetic sequence.
- 7. Insert the arithmetic mean(s) between the given terms of an arithmetic sequence. 1. 2, ___, 20 2. 5, ___, ___, 14 3. 9, ___, ___, ___, 25 4. 8, ___, ___, ___, ___, 33 5. 3, ___, ___, ___, ___, ___, 45 Arithmetic Means
- 8. 1. 2, 11, 20 𝑑 = 9 2. 5, 8, 11, 14 𝑑 = 3 3. 9, 13, 17, 21, 25 𝑑 = 4 4. 8, 13, 18, 23, 28, 33 𝑑 = 5 5. 3, 10, 17, 24, 31, 38, 45 𝑑 = 7 Arithmetic Means
- 9. Question: What is 𝑎5 in the arithmetic sequence 5, 9, 13, 17, …? Answer: It is 21. Add 4 to 17 (𝑎4) since 𝑑 = 4. How about 𝑎199? Finding the 𝑛th Term of an Arithmetic Sequence
- 10. Finding the 𝑛th Term of an Arithmetic Sequence 𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑 where: 𝑎𝑛 is the 𝑛th term / last term 𝑎1 is the first term 𝑛 is the number of terms 𝑑 is the common difference
- 11. 𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑 Question 1: What is 𝑎199 in the arithmetic sequence 5, 9, 13, 17, …? Given: 𝑎1 = 5 𝑑 = 4 𝑛 = 199 𝑎199 =? Solution: 𝑎199 = 5 + 199 − 1 (4) 𝑎199 = 5 + 198 (4) 𝑎199 = 5 + 792 𝑎199 = 𝟕𝟗𝟕 ∴ 797 is 𝒂𝟏𝟗𝟗
- 12. 𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑 Question 2: In the arithmetic sequence 5, 9, 13, 17, …, what term is 105? Given: 𝑎? = 105 𝑎1 = 5 𝑑 = 4 𝑛 =? Solution: 105 = 5 + 𝑛 − 1 (4) 105 = 5 + 4𝑛 − 4 105 − 5 + 4 = 4𝑛 104 = 4𝑛 ∴ 105 is 𝒂𝟐𝟔. 𝟐𝟔 = 𝒏
- 13. 𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑 Question 3: What is the first term of the arithmetic sequence with 𝑑 = 4 and 𝑎40 = 161? Given: 𝑎40 = 161 𝑑 = 4 𝑛 = 40 𝑎1 =? Solution: 161 = 𝑎1 + 40 − 1 (4) 161 = 𝑎1 + (39)(4) 161 = 𝑎1 + 156 161 − 156 = 𝑎1 ∴ 𝒂𝟏 is 5. 𝟓 = 𝒂𝟏
- 14. 𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑 Question 4: In an arithmetic sequence, 𝑎1 = 5 and 𝑎20 = 81, what is the common difference? Given: 𝑎1 = 5 𝑎20 = 81 𝑛 = 20 𝑑 =? Solution: 81 = 5 + 20 − 1 (𝑑) 81 = 5 + (19)(𝑑) 81 = 5 + 19𝑑 81 − 5 = 19𝑑 ∴ The 𝒅 is 4. 76 = 19𝑑 𝟒 = 𝒅
- 15. Finding the 𝑛th Term of an Arithmetic Sequence 𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑 “What is the 𝑛th term… ?” : 𝒂𝒏 = 𝒂𝟏 + 𝒏 − 𝟏 𝒅 “What term is … ?” / “How many terms … ?” : 𝒏 = 𝒂𝒏−𝒂𝟏 𝒅 + 𝟏 “What is the first term… ?” : 𝒂𝟏 = 𝒂𝒏 − 𝒏 − 𝟏 𝒅 “What is the common difference… ?”: 𝒅 = 𝒂𝒏−𝒂𝟏 𝒏−𝟏
- 16. Given: 𝑎1 = 26 052 𝑑 = 950 𝑛 = 10 Solution: 𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑 𝑎10 = 26 052 + 10 − 1 (950) 𝑎10 = 26 052 + 9 (950) 𝑎10 = 26 052 + 8 550 𝑎10 = 34 602 Thus, Jay’s salary will be ₱ 34 602. Question 5: A university hired Jay as an instructor. His starting salary is ₱26 052. Each year, he will receive a raise in the amount of ₱950. How much will be Jay’s salary during his 10th year in the university?
- 17. 1. What is the 86th term in the arithmetic sequence 8, 12, 16, 20, …? 2. In the arithmetic sequence 3, 11, 19, 27, …, what term is 387? Answer the following questions. Practice Exercises
- 18. 3. In an arithmetic sequence, the common difference is 6 and the 33rd term is 197. What is the first term of the sequence? 4. In an arithmetic sequence, the first term is 2 and the 41st term is 202. What is the common difference? Answer the following questions. Practice Exercises
- 19. 5. A spacious hall has 25 rows of seats. The last row has 94 seats. If each row contains two fewer seats than the row behind it. How many seats are there in the first row? Answer the following questions. Practice Exercises
- 20. 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.
- 21. Carl Friedrich Gauss was in the class and had the answer in a minute or two (remember no calculators in those days) 1 + 2 + 3 + 4 + 5 + . . . + 96 + 97 + 98 + 99 + 100 sum is 101 sum is 101 With 100 numbers there are 50 pairs that add up to 101. 50(101) = 5050
- 22. Arithmetic Series • is the sum of the first 𝑛 terms of an arithmetic sequence • denoted as 𝑺𝒏 Example: What is the sum of the first 5 terms of the arithmetic sequence 3, 10, 17, 24, …? 𝑺𝟓 = 𝟑 + 𝟏𝟎 + 𝟏𝟕 + 𝟐𝟒 + 𝟑𝟏 𝑺𝟓 = 𝟖𝟓
- 23. Arithmetic Series How about finding the sum of the first 86 terms of the arithmetic sequence 3, 10, 17, 24, …? 𝑺𝟖𝟔 = 𝟑 + 𝟏𝟎 + 𝟏𝟕 + 𝟐𝟒 + 𝟑𝟏 + ⋯ + 𝒂𝟖𝟔 𝑺𝟖𝟔 =?
- 24. Finding the Sum of the First 𝑛 Terms of an Arithmetic Sequence 𝑆𝑛 = 𝑛 2 2𝑎1 + 𝑛 − 1 𝑑 where: 𝑆𝑛 is the sum of the first 𝑛 term 𝑎1 is the first term 𝑛 is the number of terms 𝑑 is the common difference
- 25. 𝑆𝑛 = 𝑛 2 2𝑎1 + 𝑛 − 1 𝑑 Question 1: What is the sum of the first 86 terms of the arithmetic sequence 3, 10, 17, 24, …? Given: 𝑛 = 86 𝑎1 = 3 𝑑 = 7 Solution: 𝑆86 = 86 2 2 3 + (86 − 1)(7) 𝑆86 = 43 6 + (85)(7) 𝑆86 = 43(6 + 595) 𝑆86 = 43(601) 𝑺𝟖𝟔 = 𝟐𝟓 𝟖𝟒𝟑
- 26. Finding the Sum of the First 𝑛 Terms of an Arithmetic Sequence 𝑆𝑛 = 𝑛 2 𝑎1 + 𝑎𝑛 where: 𝑆𝑛 is the sum of the first 𝑛 term 𝑛 is the number of terms 𝑎1 is the first term 𝑎𝑛 is the last term
- 27. 𝑆𝑛 = 𝑛 2 𝑎1 + 𝑎𝑛 Question 2: What is the sum of the first 15 terms of the arithmetic sequence if the first term is 11 and the 15th term is 109? Given: 𝑎1 = 11 𝑎15 = 109 𝑛 = 15 Solution: 𝑆15 = 15 2 (11 + 109) 𝑆15 = 15 2 (120) 𝑺𝟏𝟓 = 𝟗𝟎𝟎
- 28. Given: 𝑎1 = 5 000 𝑑 = −500 𝑛 = 10 Solution: 𝑆𝑛 = 𝑛 2 2𝑎1 + 𝑛 − 1 𝑑 𝑆10 = 10 2 2 5 000 + (10 − 1)(−500) 𝑆10 = 5 10 000 + (9)(−500) 𝑆10 = 5 10 000 − 4 500 𝑆10 = 5(5 500) 𝑆10 = 𝟐𝟕 𝟓𝟎𝟎 Hence, the total prize money is ₱ 27 500. Question 3: A non-government organization will be holding a competition in which the top 10 finishers win cash prizes. The first placer will receive a cash prize of ₱ 5000, the second placer will receive ₱ 4500, the third placer will receive ₱ 4000, and so on. How much is the total of prize money to be awarded?
- 29. 1. In the arithmetic sequence 4, 10, 16, 22, 28, …, what is the sum of the first 27 terms? 2. The first term in an arithmetic sequence is 3 and the 13th term is 159. What is the sum of the first 13 terms of the sequence? Answer the following questions. Practice Exercises
- 30. 3. What is the sum of all two-digit even natural numbers? 4. An employee has a salary of ₱ 304 800 per year. The employee is promised a ₱ 300 raise each subsequent year. What is the total earning over a 10-year period? Answer the following questions. Practice Exercises
- 31. 1. What are the 9 arithmetic means between 12 and 92? 2. What is 𝑎37 in the arithmetic sequence -9, -2, 5, 12, 19, … ? Answer the following questions. Activity
- 32. 3. What is 𝑎106 in the arithmetic sequence 34, 31, 28, … ? 4. In the arithmetic sequence 2, 10, 18, 26, 34, … , what term is 242? Answer the following questions. Activity
- 33. 5. If 𝑎33 = 133 and 𝑎35 = 141, what is the first term? 6. In an arithmetic sequence, 𝑎1 = 4 and 𝑎36 = 179. What is the common difference? Answer the following questions. Activity
- 34. 7. What is the sum of the first 18 terms of the arithmetic sequence 10, 18, 26, 34, 42, … ? 8. In an arithmetic sequence, 𝑎1 = 5 and 𝑎50 = 201. What is the sum of the first 50 terms of the sequence? Answer the following questions. Activity
- 35. 9. What is the sum of all the odd integers from 1 to 99? 10.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? Answer the following questions. Activity