4.7 Arithmetic Sequences. Arithmetic sequences

Transcription

1 4.7 Arithmetic Sequences (Recursive & Explicit Formulas) Arithmetic sequences are linear functions that have a domain of positive consecutive integers in which the difference between any two consecutive terms is equal. is a list of terms separated by a common difference, (the number added to each consecutive term in an arithmetic sequence.) can be represented by formulas, either explicit or recursive, and those formulas can be used to find a certain term of the sequence. 1

2 Recursive Formula The following slides will answer these questions: What does recursive mean? What is the math formula? How do you write the formula given a situation? How do you use the formula? Definition of Recursive relating to a procedure that can repeat itself indefinitely repeated application of a rule 2

3 Recursive Formula A formula where each term is based on the term before it. Formula: A(n) = A(n-1) + d, A(1) =? For each list of numbers below, determine the next three numbers in the list. 1, 2, 3, 4, 5,,, 7, 9, 11, 13, 15,,, 10, 7, 4, 1, -2,,, 2, 4, 7, 11, 16,,, 1, -1, 2, -2, 3, -3,,, 3

4 For each list of numbers below, determine the next three numbers in the list. 1, 2, 3, 4, 5, 6, 7, 8 7, 9, 11, 13, 15,,, 10, 7, 4, 1, -2,,, 2, 4, 7, 11, 16,,, 1, -1, 2, -2, 3, -3,,, For each list of numbers below, determine the next three numbers in the list. 1, 2, 3, 4, 5, 6, 7, 8 7, 9, 11, 13, 15, 17, 19, 21 10, 7, 4, 1, -2,,, 2, 4, 7, 11, 16,,, 1, -1, 2, -2, 3, -3,,, 4

5 For each list of numbers below, determine the next three numbers in the list. 1, 2, 3, 4, 5, 6, 7, 8 7, 9, 11, 13, 15, 17, 19, 21 10, 7, 4, 1, -2, -5, -8, -11 2, 4, 7, 11, 16,,, 1, -1, 2, -2, 3, -3,,, For each list of numbers below, determine the next three numbers in the list. 1, 2, 3, 4, 5, 6, 7, 8 7, 9, 11, 13, 15, 17, 19, 21 10, 7, 4, 1, -2, -5, -8, -11 2, 4, 7, 11, 16, 22, 29, 37 1, -1, 2, -2, 3, -3,,, 5

6 For each list of numbers below, determine the next three numbers in the list. 1, 2, 3, 4, 5, 6, 7, 8 7, 9, 11, 13, 15, 17, 19, 21 10, 7, 4, 1, -2, -5, -8, -11 2, 4, 7, 11, 16, 22, 29, 37 1, -1, 2, -2, 3, -3, 4, -4, 5 These are all examples of you using the recursive formula. As long as you know the previous term and the common difference! A(n) = A(n-1) +d, A(1) =? A(n) is the nth term of the sequence A(n-1) is the previous term to nth term d is the common difference A(1) is the first term of the sequence 6

7 A(n) = A(n-1) +d, A(1) =? A(n) is the nth term of the sequence To write a recursive A(n-1) is the formula previous you term only to need nth term to identify the first term, d is the common A(1) and difference the common difference. A(1) is the first term of the sequence Example: Write the recursive formula Briana borrowed $870 from her parents for airfare to Europe. She will pay them back at the rate of $60.00 per month. Let A(n) be the amount she still owes after n months. Describe in words: Start at $870 and add -60 each time. The formula is : A(n) = A(n-1) + d, A(1) =? A(n) = A(n-1) + -60, A(1) = 870 7

8 Example: Write the recursive formula If you buy a new car, you might be advised to have an oil change after driving 1,000 miles and every 3,000 miles thereafter. The following sequence gives the mileage when oil changes are required: Describe in words: Start at 1,000 and add 3,000 each time. The formula is : A(n) = A(n-1) + d, A(1) =? 10/6/14 A(n) = A(n-1) + 3,000, A(1) = 1,000 Example: Write the recursive formula Consider the sequence generated by 2000, 2040, 2080, 2120, 2160,... Describe this sequence in words: Start at 2000 and add 40 each time. The formula is : A(n) = A(n-1) + d, A(1) =? 8

9 Using the In order to find the 4 th term, you need to find the 2 nd term, 3 rd term and then find the 4 th term. (Basically using the formula three times! using it recursively!) First Term = 2000 Second Term = = 2040 Third Term = = 2080 Fourth Term = =

10 A(1) = 2000 A(2) = = 2040 A(3) = = 2080 A(4) = = 2120 A(2) = A(2-1) +40, A(2) =? A(1)

11 A(2) = A(2-1) +40, A(2) = 2040 A(1) A(2) = A(2-1) +40, A(2) = 2040 A(3) = A(3-1) +40, A(3) =? A(2)

12 A(2) = A(2-1) +40, A(2) = 2040 A(3) = A(3-1) +40, A(3) = 2080 A(2) A(2) = A(2-1) +40, A(2) = 2040 A(3) = A(3-1) +40, A(3) = 2080 A(4) = A(4-1) +40, A(4) =? A(3)

13 A(2) = A(2-1) +40, A(2) = 2040 A(3) = A(3-1) +40, A(3) = 2080 A(4) = A(4-1) +40, A(4) = 2120 A(3) First Term = 2000 Second Term = = 2040 Third Term = = 2080 Fourth Term = =

14 A(2) = A(2-1) +40, A(2) = 2040 A(3) = A(3-1) +40, A(3) = 2080 A(4) = A(4-1) +40, A(4) = 2120 A(2) = A(2-1) +40, A(2) = 2040 Recursive: repeated A(3) = application A(3-1) +40, of a A(3) = 2080 rule. A(4) = A(4-1) +40, A(4) =

15 A(2) = A(2-1) +40, A(2) = 2040 Is there an easier way to A(3) = A(3-1) find the +40, 4 th A(3) = 2080 term? A(4) = A(4-1) +40, A(4) = 2120 Now find the 100 th Term in each sequence 1, 2, 3, 4, 5, 6, 7, 8 7, 9, 11, 13, 15, 17, 19, 21 10, 7, 4, 1, -2, -5, -8, -11 2, 4, 7, 11, 16, 22, 29, 37 1, -1, 2, -2, 3, -3, 4, -4, 5 Is the recursive formula a useful way to find the value of the 100 th term? 15

16 Recursive Formula Is the recursive formula efficient to finding the 2 nd or 3 rd term in a sequence? YES Is the recursive formula efficient in finding the 75 th term in a sequence? NO Explicit Formula The following slides will answer these questions: What does explicit mean? What is the math formula? How do you write the formula given a situation? How do you use the formula? 16

17 Definition of Explicit stated clearly and in detail leaving no room for confusion or doubt Explicit Formula: Formula where any term can be found by substituting the number of that term. Function rule that allows you to find any term, as long as you know which term you want to find A(n) = A(1) + (n 1)d 17

18 Explicit Formula: A(n) = A(1) + (n 1)d A(n) is the nth term of the sequence A(1) is the first term of the sequence d is the common difference Example: Write the explicit formula If you buy a new car, you might be advised to have an oil change after driving 1,000 miles and every 3,000 miles thereafter. At what milage would you have your 5 th oil change? Describe this sequence in words: Start at 1000 and add 3,000 each time. A(n) = A(1) + (n 1)d A(5) = (5 1) /6/14 Your 5 th oil change would be at 13,000 miles. 18

19 Got it? #3: On pg. 276, Complete Got it #3 a-d Use the Explicit formula, NOT the recursive formula. 10/6/14 Got it? # 3 ANSWERS a) A(9) = 51 b) A(9) = 119 c) A(9) = 11.3 d) A(9) = 25 10/6/14 19

20 Got it? #4: On pg. 277, Complete Got it #4 a & b 10/6/14 Got it? # 4A ANSWER A(n) = A(1) + (n 1)d = (16 1) (-1.75) = (15) (-1.75) = $73.75 * The Key word is AFTER 15 rides. 10/6/14 20

21 Got it? # 4B ANSWER = or 57 rides 10/6/14 Arithmetic Sequences Since you must know the first term and the common difference for both the explicit and recursive formula, you can write the formulas interchangeably. Try it in Got it #5 & #6 on pg /6/17 21

22 Explicit vs. Recursive An explicit formula is a formula used to find the nth term of a sequence A recursive sequence is the process in which each step of a pattern is dependent on the step or steps before it. A recursive formula is a formula used to find the next term of a sequence when the previous term is known. 22