11/6/17 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 11/6/17 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 11/6/17 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 11/6/17 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 11/6/17 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 11/6/17 For each list of numbers below, determine the next three numbers in the list. 1, 2, 3, 4, 5, 6, 7, 8 These are all examples of you 7, 9, 11, 13, 15, 17, 19, 21 using the 10, 7, 4, 1, -2, -5, -8, -11 recursive formula. As long as you 2, 4, 7, 11, 16, 22, 29, 37 know the previous 1, -1, 2, -2, 3, -3, 4, -4, 5 term and the common difference! Recursive Formula: 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 11/6/17 Recursive Formula: 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 theformula previous you term only toneed nth term to identify the first term, d is the commonA(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 11/6/17 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: 1000 4000 7000 10000 13000 16000 • Describe in words: Start at 1,000 and add 3,000 each time. • The formula is : A(n) = A(n-1) + d, A(1) = ? 10/6/14A(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) = ? A(n) = A(n-1) +40, A(1) = 2000 8 11/6/17 Using the Recursive Formula: • Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 In order to find the 4th term, you need to find the 2nd term, 3rd term and then find the 4th term. (Basically using the formula three times!—using it recursively!) Recursive Formula: Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 First Term = 2000 Second Term = 2000 + 40 = 2040 Third Term = 2040 + 40 = 2080 Fourth Term = 2080 + 40 = 2120 9 11/6/17 Recursive Formula: Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 A(1) = 2000 A(2) = 2000 + 40 = 2040 A(3) = 2040 + 40 = 2080 A(4) = 2080 + 40 = 2120 Recursive Formula: Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 A(2) = A(2-1) +40, A(2) = ? A(1) +40 2000 +40 2040 10 11/6/17 Recursive Formula: Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 A(2) = A(2-1) +40, A(2) = 2040 A(1) +40 2000 +40 2040 Recursive Formula: Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 A(2) = A(2-1) +40, A(2) = 2040 A(3) = A(3-1) +40, A(3) = ? A(2) +40 2040 +40 2080 11 11/6/17 Recursive Formula: Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 A(2) = A(2-1) +40, A(2) = 2040 A(3) = A(3-1) +40, A(3) = 2080 A(2) +40 2040 +40 2080 Recursive Formula: Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 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) +40 2080 +40 2120 12 11/6/17 Recursive Formula: Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 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) +40 2080 +40 2120 Recursive Formula: Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 First Term = 2000 Second Term = 2000 + 40 = 2040 Third Term = 2040 + 40 = 2080 Fourth Term = 2080 + 40 = 2120 13 11/6/17 Recursive Formula: Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 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 Recursive Formula: Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 A(2) = A(2-1) +40, A(2) = 2040 Recursive: repeated A(3) = applicationA(3-1) +40, of a A(3) = 2080 rule. A(4) = A(4-1) +40, A(4) = 2120 14 11/6/17 Recursive Formula: Find the 4th term of A(n) = A(n-1) +40, A(1) = 2000 A(2) = A(2-1) +40, A(2) = 2040 Is there an easier way to A(3) = A(3find-1) the +40, 4th A(3) = 2080 term? A(4) = A(4-1) +40, A(4) = 2120 Now find the 100th Term in each sequence 1, 2, 3, 4, 5, 6, 7, 8… 7, 9, 11, 13, 15, 17, 19, 21… Is the recursive 10, 7, 4, 1, -2, -5, -8, -11… formula a useful way to 2, 4, 7, 11, 16, 22, 29, 37… find the value 1, -1, 2, -2, 3, -3, 4, -4, 5… of the 100th term? 15 11/6/17 Recursive Formula Is the recursive formula efficient to finding the 2nd or 3rd term in a sequence? YES Is the recursive formula efficient in finding the 75th 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 11/6/17 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 11/6/17 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.