Unit 3 Part C: Arithmetic & Geometric Sequences Essential question: What are arithmetic & geometric sequences and when do we use them? F.IF.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

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Introduction Examples: Sequence: a list of numbers or shapes Term: the position of numbers of shapes in a sequence There are two types of sequences: infinite and finite. An infinite sequence has an infinite number of terms. A finite sequence ends and has a certain number of terms. Examples: (infinite) . (finite) Examples:

Sequences Sequences are ordered lists determined by functions. The domain of the function that generates a sequences is all natural numbers. A sequence is itself a function. We will work with two types of sequences: Arithmetic and Geometric. There are two ways sequences are generally defined – recursively and explicitly.

Arithmetic Sequences Sequences whose terms increase or decrease by the same amount. The amount of increase or decrease is called the common difference. To determine the common difference, subtract the second term from the first term. Then subtract the third term from the second term and so on. Example: What is the common difference? Now you make your own example.

You try! What is the common difference of the arithmetic sequence? What would the next term of the arithmetic sequence above be?

Geometric sequences Geometric Sequences increase or decrease by a common factor or ratio, r. Dilations are examples of geometric sequences. In this example, the circle is growing by a factor of 2, meaning it is doubling in size each term. More examples: This sequence is decaying (decreasing) at a factor of r = 1/3. This sequence is growing (increasing) at at a factor of r = 2.

You Try! 1. What is the growth or decay factor (r) of ? 2. What is the growth or decay factor (r) of

Recursive formula Recursive formula is a formula that uses the previous term to find the value of the next term. If the sequence is defined with a recursive formula, the next term is based on the term before it and the common difference. The recursive formula is known symbolically as The general form for Arithmetic Sequence is: Examples: d = 2, so we can write as

Recursive formula We can define geometric sequences recursively. We will still use to represent the previous term, but instead of adding/subtracting r we will be multiplying by r. General form for Recursive Geometric Sequence: Example: r = 5 Recursive Formula: Multiply!

Explicit formulas Explicit formulas do not require previous terms to find values of other terms. Explicitly defined sequences provide the function that will generate each term. or For arithmetic sequences, we define explicit formulas similarly to linear functions. Why do you think we can use linear functions to help us define arithmetic sequences? The formula is represented symbolically as: , d = common difference, n stays n, is the value of the first term.

Explicit formula Explicit Formulas do not require previous terms. For geometric sequences, we can write explicit formulas similarly to how we created exponential functions. Why do you think we can use exponential functions to help us define geometric sequences? The formulas is represented symbolically as: is the first term, r is the growth or decay factor and n stays n. Example: r = ¼

You try! Is this sequence an arithmetic or geometric sequence? What is the common difference or growth/decay factor?