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19-1 and 20-1: Sequences Objectives: 1. To represent sequences using recursive notation Assignment: β’ P. 305: 1-11 β’ P. 319: 1-11 β’ Challenge Problems Objective 1 YOU WILL BE ABLE TO REPRESENT SEQUENCES USING RECURSIVE NOTATION Example 1 Find the first 13 terms in the following sequence: 1, 1, 2, 3, 5, 8, β¦ This is called the Fibonacci Sequence, and in order to find the πth term, you need to know the two that came before it. This is idea behind recursion. Sequences A sequence can be expressed as a function whose domain is the set of natural numbers. A sequence is simply an ordered list of numbers. 1, 1, 2, 3, 5, 8, β¦ π Terms of the sequence 1 2 3 4 5 6 ππ 1 1 2 3 5 8 Example 2 Find the next two terms in each of the following sequences: 1. 7, 12, 17, 22, β¦ 2. 7, 35, 175, 875, β¦ Example 2 Find the next two terms in each of the following sequences: Arithmetic Sequence 1. 7, 12, 17, 22, β¦ π = common difference π1 = first term ππβ1 = term before the πth term ππ = πth term Constant difference between consecutive terms ππ+1 = term after the πth term Example 2 Find the next two terms in each of the following sequences: Geometric Sequence 2. 7, 35, 175, 875, β¦ π = common ratio π1 = first term ππβ1 = term before the πth term ππ = πth term Constant ratio between consecutive terms ππ+1 = term after the πth term Example 3 Let ππ be the πth term of the Fibonacci sequence, where π1 = 1 and π2 = 1. Write a recursive formula for ππ . Formulas A recursive formula is an equation that computes a term of a sequence based on the term or terms that precede it. ππ = ππβ1 + ππβ2 ππ = 1+ 5 π β 1β 5 π 2π 5 An explicit formula is an equation that computes any term of a sequence directly. Arithmetic Formulas A recursive formula is an equation that computes a term of a sequence based on the term or terms that precede it. ππ = ππβ1 + π ππ = π1 + π β 1 β π An explicit formula is an equation that computes any term of a sequence directly. Example 4 Write a recursive formula and an explicit formula for the arithmetic sequence with π1 = 6 and π = 3. Then find the 40th term of the sequence. Geometric Formulas A recursive formula is an equation that computes a term of a sequence based on the term or terms that precede it. ππ = π β ππβ1 ππ = π πβ1 β π1 An explicit formula is an equation that computes any term of a sequence directly. Example 5 Write a recursive formula and an explicit formula for the sequence below. Then find the 40th term of the sequence. 4096, 1024, 256, 64, β¦ Example 6 1. What type of function is a geometric sequence? 2. What type of function is an arithmetic sequence? 19-1 and 20-1: Sequences Objectives: 1. To represent sequences using recursive notation Assignment: β’ P. 305: 1-11 β’ P. 319: 1-11 β’ Challenge Problems