MDFP Mathematics and Statistics 1 ARITHMETIC Progressions

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

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

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 Using the above sequences, we have the following series: 1) ) ) Sequences and Series4

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) … (Arithmetic series) Sequences and Series5

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

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

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 = 64 8

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

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

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

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

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

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

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 d)Find the general term for this sequence. e)Write the recursive formula for this sequence. Sequences and Series15

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

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

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

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

Practice on Exercise 1 – 2 from Workbook Sequences and Series20

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