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COMMON CORE STANDARDS for MATHEMATICSFUNCTIONS: 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 MODELSF-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)1INTRO TO SEQUENCES AND SERIESGuido 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?2In the second week, he adds to his work by placing purple tiles around the fountain as shown:How many tiles did he add?

3In 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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If he continues this pattern, how many blue tiles will he need to complete his fourth week of work?

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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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What is a Sequence?

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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

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

INTRO TO SEQUENCES AND SERIES

Types of a Sequence?

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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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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)

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What is a Series?

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What is a Series?A list of numbers separated by addition signs: 1+2+4+8+....+128.

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What is a term?

14INTRO TO SEQUENCES AND SERIESWhat is a term?A specific number in a sequence or series. a1= first terma2= second terman=nth term (or last term)

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What is a sum?The addition of the terms of a sequence. S4=a1+a2+a3+a4Sn=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.16ex1. The nth term of a sequence is given by: an = n2 + 2

a) Write out the first 5 terms.

17ex 1 (continued) The nth term of a sequence is given by: an = n2 + 2 b) What is the value of the 7th term?

18ex1. (continued)The nth term of a sequence is given by: an = n2 + 2

c) Find a9.

19ex2. 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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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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Recursively defined Sequence

Find the first six terms of the famous sequence described below

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Recursively defined Sequence

Find the first six terms of the sequence described below

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