Recursive Formulas for Arithmetic and Geometric Sequences, Slides of Elementary Mathematics

Download Recursive Formulas for Arithmetic and Geometric Sequences and more Slides Elementary Mathematics in PDF only on Docsity! /22 7-8 Recursive Formulas 1 /22 Homework Read Sec 7-8. Do p448 10-21 2 7-8 Recursive Formulas /22 Use a recursive formula to list terms in a sequence. Write recursive formulas for arithmetic and geometric sequences. Notation 5 an is read “a sub n.” Reading Math an-1 (“a sub n minus 1.”) is the number immediately preceding an. (“a sub n.”) Reading Math an+1 (“a sub n plus 1.”) is the number immediately following an. (“a sub n.”) Reading Math /22 Use a recursive formula to list terms in a sequence. Write recursive formulas for arithmetic and geometric sequences. Explicit and Recursive 6 An explicit formula defines the value of a term in a sequence using the position of the term in the sequence. In other words, to find the value of a term we use a function with input value equal to the position of the term A recursive formula defines the value of a term in a sequence using previous terms. To find the value of a term we use operations on previous terms of the sequence. /22 Use a recursive formula to list terms in a sequence. Write recursive formulas for arithmetic and geometric sequences. Recursive 7 Find the first 5 terms of the sequence with The first term is given, a1 = –2. The first 5 terms are –2, –4, –10, –28, –82. a1 = –2 and an = 3an–1 + 2 for n ≥ 2. n 1 2 3 4 5 an -2 To find the next value, use the formula for an. an = 3an–1 + 2 a2 = 3a1 + 2 a2 = 3(-2) + 2 -4 a3 = 3a2 + 2 a3 = 3(-4) + 2 -10 a4 = 3a3 + 2 a4 = 3(-10) + 2 = -28 -28 a5 = 3a4 + 2 a5 = 3(-28) + 2 = -82 -82 This recursive formula tells us to find each term of the sequence by multiplying the previous term by 3 then adding 2. a3 = -10 a2 = -4 /22 Use a recursive formula to list terms in a sequence. Write recursive formulas for arithmetic and geometric sequences. Recursive Formula 10 Find the recursive formula for the sequence; -26, -22, -18, -14, -10, … The common difference is 4 and the first term is -26. The sequence is an arithmetic sequence. To find each term we simply add 4 to the previous term. an = an-1 + 4 The common difference is 1.5 and the first term is -1. The sequence is an arithmetic sequence. To find each term we simply add 1.5 to the previous term. an = an-1 + 2.5 Find the recursive formula for the sequence; -1, 0.5, 2, 3.5, … /22 Use a recursive formula to list terms in a sequence. Write recursive formulas for arithmetic and geometric sequences. Recursive Formula 11 Find the recursive formula for the sequence; 5, 1.5, -2, -5.5, … The common difference is -3.5 and the first term is 5. The sequence is an arithmetic sequence. To find each term we simply add -3.5 to the previous term. an = an-1 + -3.5 There is no common difference but there is a common ratio. The sequence is an geometric sequence. To find each term we simply multiply the previous term by 2. an = 2an-1. Find the recursive formula for the sequence; 2, 4, 8, 16, … The first term is 2. /22 Use a recursive formula to list terms in a sequence. Write recursive formulas for arithmetic and geometric sequences. Recursive Formula 12 Find the recursive formula for the sequence; 48, 24, 12, 6, … a n = 1 2 a n−1 ;a 1 = 48 There is a common ratio of . The sequence is an geometric sequence. 1 2 To find each term we simply multiply the previous term by . 1 2 Find the recursive formula for the sequence; -2, 6, -12, 24, … a n = −3a n−1 ;a 1 = −2 There is a common ratio of . The sequence is an geometric sequence. −3 To find each term we simply multiply the previous term by . −3 The first term is 48. The first term is -2. /22 Use a recursive formula to list terms in a sequence. Write recursive formulas for arithmetic and geometric sequences. Recursive Formula 15 Determine if the sequence is arithmetic, geometric, or neither. 120, 60, 20, 5, … Try a common difference 60 - 120 = -60 20 - 60 = -40 5 - 20 = -15 No common difference, thus not arithmetic Try a common ratio 60 120 = 1 2 20 60 = 1 3 5 20 = 1 4 No common ratio, thus not geometric The sequence 120, 60, 20, 5, … is neither arithmetic nor geometric. /22 Use a recursive formula to list terms in a sequence. Write recursive formulas for arithmetic and geometric sequences. Recursive Formula 16 Determine if the sequence is arithmetic, geometric, or neither. Try a common difference 40 - 120 = -80 No common difference, thus not arithmetic Try a common ratio 40 120 = 1 3 40 3 40 = 1 3 40 9 40 3 = 1 3 120, 40, 40 3 , 40 9 ,... 40 3 − 40 = −80 3 The common ratio 1 3 The sequence is geometric. First term 120, common ratio 1 3 a n = 1 3 a n−1 ;a 1 = 120 /22 Use a recursive formula to list terms in a sequence. Write recursive formulas for arithmetic and geometric sequences. Recursive Formula 17 Determine if the sequence is arithmetic, geometric, or neither. 20, 14, 8, 2, … Try a common difference 14 - 20 = -6 8 - 14 = -6 2 - 8 = -6 The common difference is -6 The sequence is arithmetic. First term 20, common difference -6 an = an-1 + -6; a1 = 20 /22 Use a recursive formula to list terms in a sequence. Write recursive formulas for arithmetic and geometric sequences. Explicit Formula 20 a1 = 9.5 a2 = 9.5 + .5 = 10 a3 = 10 + .5 = 10.5 The first term is 9.5, and the common difference is 0.5. The explicit formula is an = 9 + 0.5n. an = 9 + 0.5n 9 is the “zeroth” term, or the term before a1, and 0.5 is the common difference. Find a explicit formula for a1 = 9.5, an = an-1 + 0.5 an = a1 + (n – 1)d an = 9.5 + (n – 1)0.5 an = 9.5 + 0.5n – 0.5 an = 9 + 0.5n /22 Use a recursive formula to list terms in a sequence. Write recursive formulas for arithmetic and geometric sequences. Recursive Formula Write a recursive formula for an = 500(0.2)n–1 21 The first term is 500, and the common ratio is 0.2. The recursive formula is an = (0.2)an–1, a1 = 500. Write an explicit formula for a1 = 5, an = 2an–1 The common ratio is 2 and the first term is 5; an = 5(2)n–1 The explicit formula is; an = 5(2)n–1 /22 Use a recursive formula to list terms in a sequence. Write recursive formulas for arithmetic and geometric sequences. Using Z scores 22 an = -145 + 12n Find a explicit formula for a1 = -133, an = an-1 + 12 an = a1 + (n – 1)d an = -133 + (n – 1)12 an = -133 + 12n – 12 an = -145 + 12n How did we get -145? a1 = -133, the term before would be -145 a1 = 4(1) - 24 = -20 The first term is -20, and the common difference is 4. The recursive formula is an = an-1 + 4, a1 = -20. an = a0 + ndFind a recursive formula for an = 4n - 24