MATHEMATICS Lesson-2-Arithmetic-Sequence (1).pptx

MATHEMATICS Lesson-2-Arithmetic-Sequence (1).pptx

1. Observe . 9 x 1 = 7 9 x 2 = 18 9 x 3 = 27 9 x 4 = 36 9 x 5 = 45 9 x 6 = 54 9 x 7 = 63 9 x 8 = 72 9 x 9 = 81 9 x 10 = 90

2. R + + + + + S Arithmetic Sequence

3. ILLUSTRATION The diagram shows four patterns. 1 2 3 4 Let the length of a matchsticks equal to 2 cm, M n , represents the number of matchsticks in the n th pattern, and P n , the perimeter of the pattern.

4. 1 2 3 4 n 1 2 3 4 5 M n 7 P n 12 A. Complete the table. B. Find M n . C. Find P n .

5. 1 2 3 4 n 1 2 3 4 5 M n 7 P n 12 A. Complete the table. B. Find M n . C. Find P n . 12 17 16 20 22 24 27 28 M n = 5n + 2 P n = 4n + 8

6. Definition A sequence whose consecutive terms have a common difference is an arithmetic sequence . The sequence a 1 , a 2 , a 3 , a 4 , …, a n , is arithmetic if there is a number d such that: a 2 – a 1 = d , a 3 – a 2 = d , a 4 – a 3 = d , common difference

7. EXAMPLE The sequence 1, 4, 7, 10 and 15, 11, 7, 3, –1… are examples of arithmetic sequence. 1, 4, 7, 10 15, 11, 7, 3, –1… d = d =

8. EXAMPLE The sequence 1, 4, 7, 10 and 15, 11, 7, 3, –1… are examples of arithmetic sequence. 1, 4, 7, 10 3 3 3 15, 11, 7, 3, –1… –4 –4 –4 –4 d = 3 d = –4

9. EXAMPLE Determine if the given sequence is arithmetic or not . If it is , determine the common difference . 1.) 14, 21, 28, 35, 42 2.) 5, 10, 20, 40 3.) -5, -8, -11, -14 4.) 23, 47, 71, 95 5.) 1.2, 1.6, 2.0, 2.4 6.) 0.1, 0.01, 0.001, 0.0001 7.) , , , 3 2 10 2 17 2 24 2

10. EXAMPLE Determine if the given sequence is arithmetic or not . If it is , determine the common difference . 1.) 14, 21, 28, 35, 42 2.) 5, 10, 20, 40 3.) -5, -8, -11, -14 4.) 23, 47, 71, 95 5.) 1.2, 1.6, 2.0, 2.4 6.) 0.1, 0.01, 0.001, 0.0001 7.) , , , 3 2 10 2 17 2 24 2 arithmetic d = 7 not arithmetic d = -3 arithmetic d = 24 arithmetic d = 0.4 not arithmetic d = 7 2

11. Rule The n th term, a n , of an arithmetic sequence with the first term, a 1 , and common difference, d , is given by Formula for n th Term of an Arithmetic Sequence a n = a 1 + (n – 1) d

12. Rule Given the sequence: 5, 9, 13, 17, 21, ... Step 1: The sequence is increasing by 4 each time. Step 2: The common difference is d =4 . Step 3: The first term is a 1 ​ =5 . Step 4: The general term formula is : a n= 5 + (n− 1)⋅ 4 Step 5: Substitute n into the formula : a n =5+4n−4 = 4 n +1 . So, the general term of the sequence is 4 n +1 STEPS IN DETERMINING GENERAL TERM a n = a 1 + (n – 1) d

13. Example Write a formula for the nth term of the given arithmetic sequence. a. 12, 19, 26, 33, 40, … b. 9, 1, -7, -15, -23, … a n = a 1 + (n – 1)d a n = 7n + 5 a n = 17 – 8n Solution: a. d = 7 a n = a 1 + (n – 1)d a n = 12 + (n – 1)(7) a n = 12 + 7n – 7 b. d = –8 a n = a 1 + (n – 1)d a n = 9 + (n – 1)(–8) a n = 9 – 8n + 8

14. Example Find the 15 th term of the arithmetic sequence 18, 22, 26, 30, 34, … a n = a 1 + (n – 1)d a 15 = 18 + (15 – 1)(4) a 15 = 18 + (14)(4) a 15 = 18 + 56 a 15 = 74 the 15 th term is 74. Solution:

15. Example In the arithmetic sequence 8, 14, 20, 26, 32, … , which term is 122? a n = a 1 + (n – 1)d 122 = 8 + (n – 1)(6) 114 = 6n – 6 120 = 6n n = 20 122 is the 20 th term. Solution: a n = 122 a 1 = 8 d = 6 n = ?

16. Its Your Turn Find the 11 th term of the sequence, 17, 25, 33, 41, 49, …

17. Its Your Turn Find the 11 th term of the sequence, 17, 25, 33, 41, 49, … a n = a 1 + (n – 1)d a 11 = 17 + (11 – 1)(8) a 11 = 17 + 10(8) The 11 th term is 97. a 11 = 17 + 80 a 11 = 97

18. Its Your Turn Find the 15 th term of the sequence whose first term is 5 and has common difference, –2.

19. Its Your Turn Find the 15 th term of the sequence whose first term is 5 and has common difference, –2. a n = a 1 + (n – 1)d a 15 = 5 + (15 – 1)(–2) a 15 = 5 + 14(–2) The 15 th term is –23. a 15 = 5 – 28 a 15 = –23

20. Let’s Try! Find the eighteenth term of the arithmetic sequence whose first term is 11 and whose seventh term is 59.

21. WRITTEN WORKS 1.2 In the arithmetic sequence 9,11,13,15,17, … find: a. the general rule or nth term, a n b. find its 12 th term, and c. which term is 47?