I can identify and extend patterns in sequences and represent sequences using function notation. 4.7 Arithmetic Sequences.

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I can identify and extend patterns in sequences and represent sequences using function notation. 4.7 Arithmetic Sequences

Sequences Ordered list of numbers that often form a pattern. Each number in that list a term of a sequence.

Extending a sequence Find the next 2 terms: 5, 8, 11, 14, … 2.5, 5, 10, 20, …

Arithmetic Sequence The difference between consecutive terms is constant. The difference is called a common difference.

Identifying Arithmetic sequences Tell whether the following sequences are arithmetic: 3, 8, 13, 18, … 6, 9, 13, 17, …

Recursive formula A function rule that relates each term of a sequence to the term before it. With exception to the first term EX: 7, 11, 15, 19, … The first term, A(1) = 7 The common difference is 4 Next, A(2) = A(1) + 4 = 7 + 4 = 11 A(3) = A(2) + 4 = 11 + 4 = 15 A(4) = A(3) + 4 = 15 + 4 = 19 So, A(n) = A(n – 1) + 4; A(1) = 7 is the recursive formula for this sequence.

General formula A general rule for writing a recursive formula for an arithmetic sequence is: A(n) = A(n – 1) + d n represents the term number d represents the common difference You must also include the starting value, A(1)

You try! Find the 9th term using a recursive formula: 3, 9, 15, 21, … A(n) = A(n – 1) + 6; A(1) = 3 A(5) = A(4) + 6 = 21 + 6 = 27 A(6) = A(5) + 6 = 27 + 6 = 33 A(7) = A(6) + 6 = 33 + 6 = 39 A(8) = A(7) + 6 = 39 + 6 = 45 A(9) = A(8) + 6 = 45 + 6 = 51

Practice Write a recursive rule for the following: 70, 77, 84, 91, … A(n) = A(n – 1) + 7; A(1) = 70 In order to find the 8th term, you need to extend the pattern To find A(8) I need to know A(7) To find A(7) I need to know A(6) and so on… I cannot simply plug in a term number into a recursive formula.

For an arithmetic sequence, the formula is: Explicit formula A function rule that relates each term of a sequence to the term number. You can plug it in and find a specific term. For an arithmetic sequence, the formula is:

Try it! Write an explicit formula: A(n) = 200 + (n – 1)(10) Find the value of the 12th term A(12) = 200 + (12 – 1)(10) A(12) = 200 + (11)(10) = 200 + 110 = 310

Explicit from recursive Here is a recursive formula: A(n) = A(n – 1) + 12; A(1) = 19 How can I make it explicit? I know A(1) = 19 and d = 12 A(n) = 19 + (n – 1)(12)

Recursive from explicit Here is an explicit formula: A(n) = 32 + (n – 1)(22) What do we know? A(1) = 32 and d = 22 So, A(n) = A(n – 1) + 22; A(1) = 32

Assignment ODDS P.279 #9-49