1. MDFP Mathematics and Statistics 1 ARITHMETIC Progressions

2. MDFP – M&S 1 AP Sequences and Series Lesson 21

3. 1. Definition and Example 2. Arithmetic Progressions 1.Definition, first term and common difference 2.Recursive formula 3.T n = n th term 4.Sum of the first n terms 3. EXAMPLES 4. Practice – E1 - 2 from Workbook 5. Conclusion Sequences and Series3

4. Def : A sequence is a number pattern in a definite order following a certain rule. Examples of sequences: 1) 1, 2, 3, 4, 5, 6, 7,... add 1 to the preceding term 2) 2, 4, 7, 11, 16, 22, 29,... add 2 to the preceding term, add 3 to the next term, etc 3) 1, 1, 2, 3, 5, 8, 13, 21, 34,... add the two preceding terms together Sequences are usually denoted by: T 1, T 2, T 3, T 4,... Def : A series is a sum of terms in a sequence. It can be denoted by T 1 + T 2 + T 3 + T 4 +... Using the above sequences, we have the following series: 1) 1 + 2 + 3 + 4 + 5 + 6 + 7 +... 2) 2 + 4 + 7 + 11 + 16 + 22 + 29+... 3) 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 +... Sequences and Series4

5. Def : An arithmetic progression (commonly abbreviated to A.P. ) is a sequence in which each term (except the first term) is obtained from the previous term by adding a constant known as the common difference. An arithmetic series is formed by the addition of the terms in an arithmetic progression. a, a + d, a + 2d, a + 3d,..., a + (n – 1)d  a = first term  d = common difference e.g. 1, 2, 3, 4… (Arithmetic sequence) 1 + 2 + 3 + 4… (Arithmetic series) Sequences and Series5

6. Examples: 1)  Arithmetic sequence: 1, 2, 3, 4…  Arithmetic series: 1 + 2 + 3 + 4… 2)  Arithmetic sequence: 2, 4, 6, 8, 10  Arithmetic series: 2 + 4 + 6 + 8 + 10 3)  Arithmetic Sequence: -8, -3, 2, 7  Arithmetic Series: -8 + -3 + 2 + 7 Sequences and Series6

7. Recursive Formula and n th term T n+1 = T n + dand T 1 = a T n = n th term = a + (n – 1)d Sum of first n terms of an A. P.: S n = n/2 [2a + (n - 1) d] or S n = n/2 [ first term + last term] = n/2 [ a + l] Where l is the last term Sequences and Series7

8. Example: S n is the symbol used to represent the first ‘n’ terms of a series. Given the sequence 1, 11, 21, 31, 41, 51, 61, 71, … find S 4 We add the first four terms 1 + 11 + 21 + 31 = 64 8

9. Find S 8 of the arithmetic sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36 9

10. What if we wanted to find S 100 for the sequence in the last example. It would be a pain to have to list all the terms and try to add them up. We need to use a formula ! 10

11. a = first term l = last term d = common difference Arithmetic Series = S n = n/2 (a + l)

12. Example 1 Find the sum of the first 10 terms of the arithmetic series with T 1 = 6 and T 10 =51 S n = n / 2 (a + l), OR S n = n / 2 (T 1 + T n ) S 10 = 10/2 (6 + 51) = 5(57) = 285 Sequences and Series12

13. Example 2 Find the sum of the first 50 terms of an arithmetic series with a = 28 and d = - 4 We need to know n, T 1, and T 50. n = 50, T 1 = 28, T 50 = ?? We have to find it!!! Use the formula: T n = T 1 + (n – 1)d Sequences and Series13

14. Example 2 T 50 = 28 + (-4)(50 - 1) = 28 + -4(49) = 28 + -196 = -168 So n = 50, T 1 = 28, and T 50 =-168 S 50 = 50/2 (28 + -168) = 25(-140) = -3500 Sequences and Series14

15. Example 3 An A.P. is defined as 3, 11, 19, 27, … a)Find the thirtieth term of this sequence. b)Is 430 a term in this sequence. c)What is the lowest valued term over 1000. d)Find the general term for this sequence. e)Write the recursive formula for this sequence. Sequences and Series15

16. Example 3 T 1 T 2 T 3 T 4 A.P. is defined as 3, 11, 19, 27, … a) d = 11 – 3 = 8 Double check 19 – 11 = 8 a = 3 d = 8 Using formula T 30 = a + (n – 1)d = 3 + (30 – 1)8 = 235 Sequences and Series16

17. Example 3 b)If 430 is a term of this sequence 430 = 3 + (n – 1)8 430 = 3 + 8n - 8 8n = 430 – 3 + 8 8n = 435 and n = 54.375 which is not a counting number 430 is not a term of this sequence Sequences and Series17

18. Example 3 c)If 1000 was a term of this sequence 1000 = 3 + (n – 1)8 1000 = 3 + 8n - 8 8n = 1000 – 3 + 8 8n = 1005 and n = 125.675 which is not a counting number That means that the next term which is 126 is the lowest valued term over 1000. T 126 = 3 + (126 – 1)8 = 3 + 125x8 = 1003 Sequences and Series18

19. Example 3 d)The general term of a A.P. is T n = a + (n - 1)d = 3 + (n - 1)8 = 8n – 5 e)The recursive formula for a A.P. is T n+1 = T n + d = T n + 8 Sequences and Series19

20. Practice on Exercise 1 – 2 from Workbook Sequences and Series20

21. What did you learn today? Why do you need to learn about AP Sequences and Series ? Assignment: Any work not completed during class must be completed for homework Sequences and Series21