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COMMON CORE STANDARDS for MATHEMATICS FUNCTIONS: INTERPRETING FUNCTIONS (F-IF) F-IF3. Recognize that sequences are functions, sometimes defined recursively. Whose domain is a subset of the integers. FUNCTIONS: BUILDING FUNCTIONS (F-BF) F-BF2. Write an arithmetic and geometric sequences both recursively and with explicit formula, use them to model situations and translate between the two forms. FUNCTIONS: LINEAR, QUADRATIC, AND EXPONENTIAL MODELS F-LE 2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or to input-output pairs (include reading from a table) INTRO TO SEQUENCES AND SERIES Guido wants to create a tile mosaic around the Ram-Fountain. In the first week he begins his work by placing red tiles around the fountain as shown: How many tiles did he add? Figure 1 In the second week, he adds to his work by placing purple tiles around the fountain as shown: How many tiles did he add? Figure 2 In the third week, he adds to his work by placing green tiles around the fountain as shown: How many tiles did he add? Figure 3 INTRO TO SEQUENCES AND SERIES If he continues this pattern, how many blue tiles will he need to complete his fourth week of work? INTRO TO SEQUENCES AND SERIES In the 10th week, how many tiles would you expect him to add. How many total are around the fountain? Explain how you arrived at this answer. INTRO TO SEQUENCES AND SERIES What is a “Sequence”? INTRO TO SEQUENCES AND SERIES What is an “Infinite Sequence”? An infinite sequence is a function whose domain is the set of positive integers. The function values a1, a2, a3, a4, a5, a6, a7. . . Are the terms of the sequence. If the domain of a function consists of the first n positive integers only, the sequence is a finite sequence A list of numbers separated by commas: 1, 2, 4, 8...., 128……… INTRO TO SEQUENCES AND SERIES Types of a “Sequence”? INTRO TO SEQUENCES AND SERIES Types of a “Sequence”? Arithmetic: a sequence of numbers that has a common difference (d). EX: 1, 3, 5, 7 the common difference is 2. (each term is arrived at through addition) INTRO TO SEQUENCES AND SERIES Types of a “Sequence”? Arithmetic: a sequence of numbers that has a common difference (d). EX: 1, 3, 5, 7 the common difference is 2. (each term is arrived at through addition) Geometric: a sequence of numbers that has a common ratio (r). EX: 3, 12, 48, 192 the common ratio is 4. (each term is arrived at through multiplication) INTRO TO SEQUENCES AND SERIES What is a “Series”? INTRO TO SEQUENCES AND SERIES What is a “Series”? A list of numbers separated by addition signs: 1+2+4+8+....+128. INTRO TO SEQUENCES AND SERIES What is a “term”? INTRO TO SEQUENCES AND SERIES What is a “term”? A specific number in a sequence or series. a1= first term a2= second term an=nth term (or last term) INTRO TO SEQUENCES AND SERIES What is a “sum”? The addition of the terms of a sequence. S4=a1+a2+a3+a4 Sn=a1+a2+a3+...+an ***The difference between S3 and “Writing a series of the first 3 terms” is that S3 asks you to add the terms. ex1. The nth term of a sequence is given by: an = n 2 + 2 a) Write out the first 5 terms. ex 1 (continued) The nth term of a sequence is given by: an = n 2 + 2 b) What is the value of the 7th term? ex1. (continued) The nth term of a sequence is given by: an = n 2 + 2 c) Find a9. ex2. The nth term of a sequence is given by: an = 4(n + 2)(n – 1) Use the table function of the graphing utility on your calculator to write out the first 5 terms. INTRO TO SEQUENCES AND SERIES What is a “Recursively defined Sequence”? A sequence in which calculating each term is based on the value of the term before. INTRO TO SEQUENCES AND SERIES Recursively defined Sequence Find the first six terms of the “famous” sequence described below a0 1, a1 1, ak ak 2 ak 1 INTRO TO SEQUENCES AND SERIES Recursively defined Sequence Find the first six terms of the sequence described below a1 15, ak 1 ak 3