11.2 & 11.3: Sequences What is now proven was once only imagined. William Blake.

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Presentation transcript:

11.2 & 11.3: Sequences What is now proven was once only imagined. William Blake

Sequence A sequence can be define as a function whose domain consists of consecutive positive integers.

Two Types of Sequences Arithmetic Sequences Geometric Sequences

Definitions Arithmetic Sequence: A sequence of numbers in which each term except the first term is the result of adding or subtracting the same number to the preceding term. Geometric Sequence: A sequence of numbers in which each term except the first term is the result of multiplying or dividing the same number to the preceding term.

Examples Arithmetic Sequence: 3,7, 11,15 Geometric Sequence: 5, 10,20,  x

Recursive Rules A n =A n-1 +d A n = A n-1 *r Arithmetic Geometric These rules define sequences using prior terms. Think about the Fibonacci Sequence.

Explicit Rules In an arithmetic sequence with first term t 1, and common difference d, the nth (or general) term is given by t n = t 1 + (n-1)d (p. 507) In a geometric sequence with first term t 1 and common ratio r, the nth (or general) term is given by t n = t 1 * r n-1 (p. 510)

Examples Find t 17 for the following arithmetic sequence: 5, 8, 11, 14,…… Find t 6 for the following geometric sequence: 3, -12, 48,…. Find t 23 for the arithmetic sequence in which t 2 = 4 and t 5 = 22. Find t 7 for the geometric sequence in which t 2 = 24 and t 5 = 3. Answer: 53 Answer: Answer: 130

Arithmetic Mean(s) The term(s) between two given terms of an arithmetic sequence. 10, 1210, 22 Three arithmetic means The arithmetic mean Geometric Mean(s) The term(s) between two given terms of an geometric sequence. 1, 16 Three geometric means The arithmetic mean 13, 16, 19,16, 2, 4, 8,8,

Examples Geometric Mean: 6Arithmetic Mean: , 22, 29, 36, 43, 50

Examples Find the equation for the following sequence: 5, 8, 11, 14,…… Find the equation for the following sequence: 3, -12, 48,…. Find the equation for the arithmetic sequence in which t 2 = 4 and t 5 = 22. Find the equation for the geometric sequence in which t 2 = 24 and t 5 = 3. Y = 3x + 2 or A n =3+ A n-1 Y=(-1) x +1 (.75)4 x or a n =4a n-1 Y = 6x -8 or A n =6+ A n-1 Y= (96)(.5) x or a n =(.5)a n-1