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

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7-8 Recursive Formulas

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Homework

Read Sec 7-8.

Do p448 10-

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7-8 Recursive Formulas

Write recursive formulas for arithmetic and geometric sequences. Recursive Formula 4 a 1 ,^ a 2 ,^ a 3 ,^ a 4 , …, an-2, an-1, an, an+1, an+2, … A sequence is an ordered set of numbers. Each number in the sequence is a term of the sequence. A sequence may be an infinite sequence that continues without end, such as the natural numbers, or a finite sequence that has a limited number of terms, such as {1, 2, 3, 4}. Values in the domain are called term numbers and are represented by n. Instead of function notation, such as a(n), sequence values are written by using subscripts. The first term is a 1 , the second term is a 2 , and the nth term is a n

You can think of a sequence as a function with sequential natural numbers as the domain and the terms of the sequence as the range.

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

Write recursive formulas for arithmetic and geometric sequences. Recursive 7 Find the first 5 terms of the sequence with The first term is given, a 1 = –2. The first 5 terms are –2, –4, –10, –28, –82. **a 1 = –2 and a n = 3a n–

• 2 for n ≥ 2.** n 1 2 3 4 5 an - To find the next value, use the formula for an. a n = 3 a n–

a 2 = 3 a 1

a 2

a 3 = 3a 2

a 3

a 4 = 3 a 3

a 4

a 5 = 3 a 4

a 5

This recursive formula tells us to find each term of the sequence by multiplying the previous term by 3 then adding 2. a 3 = - a 2

Write recursive formulas for arithmetic and geometric sequences. Linear Sequence 8 An arithmetic sequence is a sequence where the difference between successive terms is a constant. The difference is the “ common difference ” and the sequence is linear. n 1 2 3 4 5 an 2 7 12 17 22 +1 +1 +1 + +5 +5 +5 + The common difference is 5 and the sequence is an arithmetic sequence. To find each term we simply add 5 to the previous term. (^) a n = a n-1 +^^5

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. a n = a n-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. a n = a n-1 + 2. Find the recursive formula for the sequence; -1, 0.5, 2, 3.5, …

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. a n = a n-1 + -3. 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. a n = 2 a n-1. Find the recursive formula for the sequence; 2, 4, 8, 16, … The first term is 2.

Write recursive formulas for arithmetic and geometric sequences. Recursive Formula The recursive formula for an arithmetic sequence is 13 an = an – 1 + d

an is the nth term

an – 1 is the term immediately before the nth term. d is the common difference The recursive formula for a geometric sequence is (^) a n =^ ran – 1

an is the nth term

an – 1 is the term immediately before the nth term. r is the common ratio

Write recursive formulas for arithmetic and geometric sequences. Recursive Formula From your book 14

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 = - 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

a

n− 1

; a

1

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 = - The common difference is - The sequence is arithmetic. First term 20, common difference - a n = a n-1 + -6; a 1 = 20

Write recursive formulas for arithmetic and geometric sequences. Continuity Correction Find a recursive formula for a n = -2n - 1 19 a 1 = -2(1) - 1 = - a 2 = -2(2) - 1 = - a 3 = -2(3) - 1 = - The first term is -3, and the common difference is -2. The recursive formula is a n = a n-1 - 2. a n = a 0 + nd a n = a n-1 + - -1 is the “zeroth” term, or the term before a 1 , and -2 is the common difference.

Write recursive formulas for arithmetic and geometric sequences. Explicit Formula 20 a 1 = 9. a 2 = 9.5 + .5 = 10 a 3 = 10 + .5 = 10. The first term is 9.5, and the common difference is 0.5. The explicit formula is a n = 9 + 0.5 n. a n = 9 + 0.5n 9 is the “zeroth” term, or the term before a 1 , and 0.5 is the common difference. Find a explicit formula for a 1 = 9.5, a n = a n-1 + 0.

a n = a 1 + ( n – 1) d

a n = 9.5 + ( n – 1) 0.

a n = 9.5 + 0.5n – 0.

a n = 9 + 0.5n