Warm-up: Give the next 2 terms in each sequence: 2, 8, 14, 20, __, __ 72, 36, 18, __, __ 1, 1, 2, 3, 5, __, __ 26, 32 9, 4.5 8, 13

HW Key: p. 779 2) 6 4) -4 8) -2, 10, 22, 34, 46 14) a8 = -31; an = -4n + 1 16) a8 = -3; an = -2n + 13 18) a8 = -15; an = -3n + 9 20) a8 = 29; an = -3n + 53 42) an = -n + 2; D: {1,2,3,4,5} R: {-3,-2,-1,0,1}

Arithmetic Sequences: Recursive Formula Chapter 11 Arithmetic Sequences: Recursive Formula

Objectives & HW The students will be able to write arithmetic sequences in recursive form. HW: Write the recursive formula for: p. 779: 14, 16, 18, 20

Recursive Form of a Sequence Ex: Define the sequence {-5, -1, 3, 7, . . .} recursively. The first term is a1 = -5. We get the next term by adding 4 to the term before it. Hence, an = an-1 + 4. The recursive definition is always made up of two parts: a1 = -5 an = an-1 + 4 for n > 1

Students Try! Ex: Find the first 5 terms of the given sequence: a1 = 6 an = 3an-1 + 2 for n > 1 6, 20, 62, 188, 566

Recursive Formula for the nth term of a Sequence: In an arithmetic sequence: a1 = # an = an-1 + d where d = common difference.

Recursive definition is a set of statements that specifies one or more initial terms, and defines the nth term as an expression involving the previous term or terms. Explicit definition: A formula in which each term is defined as an expression involving n, where n is a natural number. The expression does not involve previous terms.

Ex: In an arithmetic sequence, a 5 = 11 and a12 = 39 Ex: In an arithmetic sequence, a 5 = 11 and a12 = 39. Find a recursive formula for an. The recursive formula is: a1 = -5 an = an – 1 + 4, for n > 1

Applications: Going backwards Elliott borrowed $370 from his parents and he will pay them at the rate of $60 per month. How long will it take him to pay his parents back? Generate the sequence. Find a1 and d. Write an equation (explicit & recursive). Solve and answer the problem. Note that a0 = $370 and a1 = $310 . Answer is 7 months and he will pay $10 on the 7th month.

Applications: You try!!! Suppose you participate in a bike-a-thon for charity. The charity starts with $1100 in donations. Each participant must at least raise $35 in pledges. What is the minimum amount of money raised if there are 75 participants? (i.e. What is the 75th term?) (Be careful when identifying the first term in the sequence!) Note that a0 = $1100 and a1 = $1135.

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