Arithmetic-Sequence-Quarter 1 week 2 Grade 10

Arithmetic Sequence for Grade 10

1. nth term of an arithmetic sequence arithmetic means arithmetic series

2. At the end of the learning session, the student is expected to: - determine the nth term of an arithmetic sequence; - determine the means of an arithmetic sequence; - find the sum of the first n terms of an arithmetic sequence. Objectives

3. Determine if the given sequence is arithmetic or not. If it is an arithmetic sequence, write the “AS” followed by the common difference and if not, write the word “NOT”. 1. 2, 4, 6, 8, 10, … 2. 64, 52, 40, 28, 16, … 3. , , , , , … 4. 0, 1, 1, 2, 3,… 5. 3, , , 1, , …  

4. Lesson 1 Nth Term of an Arithmetic Sequence The nth term or general rule of an arithmetic sequence is described by the equation a ₙ = a ₁ + (n – 1)d where a ₙ is the last term, a ₁ is the first term, n is the number of terms, and d is the common difference

5. Example 1. What is the 15 th term of the sequence 9,13, 17, 21, 25, …? Answer: Given: a ₁ = 9 n=15 d = 4 Unknown : a₁₅ Solution: a ₙ = a ₁ + (n – 1)d a ₁₅ = 9 + (15 – 1)(4) = 9 + (14)(4) = 9 + 56 = 65 Therefore, the 15 th term of the arithmetic sequence 9, 13, 17, 25, … is 65.

6. Example 2 : What is the first term of an arithmetic sequence whose 17 th term and common difference are 148 and 7, respectively? Answer: Given: a₁₇ = 148 n=17 d = 7 Unknown : a₁ Solution: aₙ = a₁ + (n–1)d a ₁₇ = a₁ + (17–1)(7) 148 = a₁ + (16)(7) 148 = a₁ + 112 148–112 = a₁ 36 = a₁ Therefore, the first term of an arithmetic sequence with 17 th term of 148 and a common difference of 7 is 36.

7. Example 3 : The 11 th and 25 th terms of an arithmetic sequence are 130 and 74, respectively. Find the common difference. Answer: Given: a ₁₁ = 130 a ₂₅ = 74 Unknown : d Solution: a ₙ = a ₁ + (n – 1)d a ₙ = a ₖ + (n –k )d a₂₅ = a ₁₁ + (25 – 11)d 74 = 130 + (14)d 74 – 130 = 14d – 56 = 14d – 4 = d

8. Example 4: Which term of the arithmetic sequence 83, 89, 95, 101, … is 2021? Answer: Given: a ₙ = 2 021 a₁=83 d = 6 Unknown : n Solution: a ₙ = a ₁ + (n – 1)d 2 021 = 83 + (n – 1)(6) 2021 = 83 + 6n – 6 2021 – 83 + 6 = 6n 1 944 = 6n 324 = n  

9. Lesson 2 Arithmetic means are the terms between two nonconsecutive terms of an arithmetic sequence. Arithmetic Means

10. Arithmetic Means Example 5: a. Insert 5 arithmetic means between 17 and 65. b. Insert 3 arithmetic means between 93 and 41. c. Insert an arithmetic mean between 42 and 18. To determine the arithmetic means, we must identify first the common difference:

11. Arithmetic Means Insert 5 arithmetic means between 17 and 65 a ₙ = a ₁ + (n – 1)d a₇ = a₁ + (7 – 1)d 65 = 17 + 6d 65 -17 = 6d 48 = 6d 8 = d 17, ____, ____, ____, ____, ____, 65 a ₁ a ₂ a ₃ a ₄ a ₅ a ₆ a ₇ 25 33 49 41 57

12. Arithmetic Means Insert 3 arithmetic means between 93 and 41. a ₙ = a ₁ + (n – 1)d a₅ = a₁ + (5–1)d 41 = 93 + 4d 41–93 = 4d –52 = 4d –13 = d 93 , ____, ____, ____, 41 a ₁ a ₂ a ₃ a ₄ a ₅ 80 67 54

13. Arithmetic Means Insert an arithmetic mean between 42 and 18. 42 , ____, 18 a ₁ a ₂ a ₃ An arithmetic mean is also called as average. Instead of finding the common difference to determine the arithmetic mean, we may use a₂ = . Therefore, the arithmetic mean or average of 42 and 18 is 30.   30

14. Arithmetic Series Lesson 3 Arithmetic Sequence 1, 4, 7, 10, 13, 16, … , 94 Arithmetic Series 1 + 4 + 7 + 10 + 13 + 16 + … +94 Arithmetic series is the sum of the terms in an arithmetic sequence. Formula: Sₙ = (a₁ + aₙ)   Sₙ = [2a₁ + (n–1)d]   or

15. Example 6: Find the sum of 1 + 4 + 7 + 10 + 13 + 16 + … + 94. Arithmetic Serie s Before solving for the sum of the terms, we need to determine the number of terms in the sequence using the general rule of an arithmetic sequence.

16. Example 6: Find the sum of 1 + 4 + 7 + 10 + 13 + 16 + … + 94. Arithmetic Serie s aₙ = a₁ + (n – 1)d 94 = 1 + (n – 1)(3) 94 = 1 + 3n – 3 94 – 1 +3 = 3n 96 = 3n 32 = n  Sₙ = (a₁ + aₙ) Sₙ = (1 + 94) Sₙ = 16 (95) Sₙ = 1 520   To solve for the sum: T he sum of 1 + 4 + 7 + 10 + 13 + 16 + … + 94 is 1 520.

17. Example 7: Find the sum of first 87 terms of the arithmetic sequence 148, 143, 138, 133, 128, …. Arithmetic Serie s Since we can identify the number of terms, the first term and the common difference but not the last term, it is advisable to use the formula Sₙ = [2a₁ + (n–1)d] .  

18. Example 7: Find the sum of first 87 terms of the arithmetic sequence 148, 143, 138, 133, 128, …. Arithmetic Serie s Sₙ = [2a₁ + (n–1)d] S₈₇ = [2(148) + (87–1)( –5)] S₈₇ = [296 + 86(–5)]   S₈₇ = (296 – 430) S₈₇ = (– 134) S₈₇ = – 5 829   Thus, the sum of the first 87 terms of an arithmetic sequence 148, 143, 138, 133, 128, … is – 5 829.

19. Example 8: Find the sum of all positive multiple of 7 integers between 30 and 150. Arithmetic Serie s The multiples of 7 between 30 to 150 are 35, 42, 49, 56, …, 147. The common difference is 7, the first term is 35 and the last term is 147. Thus, we need to determine the number of terms before finding the sum.

20. Example 8: Find the sum of all positive multiple of 7 integers between 30 to 150. Arithmetic Serie s aₙ = a₁ + (n-1)d 147 = 35 + (n-1)(7) 147 = 35 + 7n – 7 147- 35 + 7 = 7n 119 = 7n 17 = n To s olve for the n : There are 17 positive multiples of 7 integers between 30 to 150.

21. Example 8: Find the sum of all positive multiple of 7 integers between 30 to 150. Arithmetic Serie s Sₙ = (a₁ + aₙ) S₁₇ = (35 + 147) S₁₇ = (182) S ₁₇ = 1 547   To s olve for the sum: There sum of all positive multiples of 7 integers between 30 to 150 is 1 547.

22. A ny clarifications? 22

23. 1. What is the arithmetic mean of 2 and 32? 2. Insert 3 arithmetic means between -4 and 8 3. If the 3rd term of an arithmetic sequence is 13 and the 9th term is 37, what is the common difference? 4. How many terms are there in the sequence 1, 4, 7, 10, …, 682? 5 . Find the sum of odd integers between 10 and 200 Answer each of the following:

24. Answer each of the following: 1 . Insert 3 arithmetic means between -16 and -8. 2 . What is the average of 1 and ? 3. If the 4th term of an arithmetic sequence is 24 and the 12th term is 56, what is the first term? 4 . Find the sum of all positive multiples of 8 less than 2021 5 . (-45) + (-38) + (-31) + (-24)+(-17)+ . . . + 26 th term  

25. Answer each of the following: 1 . What is the average of n and p ? 2. Find the 31st term of the arithmetic sequence 7, 25/4, 11/2, . . .. 3. The 3rd term and 15th term of an arithmetic sequence are 130 and 22, respectively. Find the 32nd term. 4. Find the value of x so that 4 – 3x, 1 – x, 3 + 2x forms an arithmetic sequence 5 . Find the sum of multiples of 6 from 45 to 170

26. Thank you!

27. Let’s Enrich: 1. -14, -12, -10 2. 3. 12 4. 255 024 5 . 130   Let’s Practice: 1. 17 2. -1, 2, 5 3. 4 4. 228 5. 9 975 Let’s Think : 1. 2 . 3 . 131 4 . 5 5 . 2 268