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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)

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**How many tiles did he add?**

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?

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**How many tiles did he add?**

In the second week, he adds to his work by placing purple tiles around the fountain as shown: How many tiles did he add?

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**How many tiles did he add?**

In the third week, he adds to his work by placing green tiles around the fountain as shown: How many tiles did he add?

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**INTRO TO SEQUENCES AND SERIES**

If he continues this pattern, how many blue tiles will he need to complete his fourth week of work?

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**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.

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**INTRO TO SEQUENCES AND SERIES**

What is a “Sequence”?

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**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 What is a “Sequence”? A list of numbers separated by commas: 1, 2, 4, 8...., 128.

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**INTRO TO SEQUENCES AND SERIES**

Types of a “Sequence”?

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**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)

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**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, the common ratio is 4. (each term is arrived at through multiplication)

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**INTRO TO SEQUENCES AND SERIES**

What is a “Series”?

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**INTRO TO SEQUENCES AND SERIES**

What is a “Series”? A list of numbers separated by addition signs:

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**INTRO TO SEQUENCES AND SERIES**

What is a “term”?

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**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)

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**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.

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ex1. The nth term of a sequence is given by: an = n2 + 2 a) Write out the first 5 terms.

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ex 1 (continued) The nth term of a sequence is given by: an = n2 + 2 b) What is the value of the 7th term?

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ex1. (continued) The nth term of a sequence is given by: an = n2 + 2 c) Find a9.

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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.

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**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.

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**INTRO TO SEQUENCES AND SERIES**

Recursively defined Sequence Find the first six terms of the “famous” sequence described below

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**INTRO TO SEQUENCES AND SERIES**

Recursively defined Sequence Find the first six terms of the sequence described below

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Arithmetic vs. Geometric Sequences and how to write their formulas

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