advertisement

P.o.D. 3⁄ 4 1.) Evaluate 256 3 2.) Simplify √64𝑥 3 𝑦 6 𝑧 with rational exponents. 3.) Solve 9 = 3𝑥 − 2 4.) Solve log 𝑥 243 = 5 5.) Solve 3.4𝑥 = 82.3 12-1: Sequences and Series Learning Target: I will be able to use sequence notation to write the terms of sequences; use summation notation to write sums; find the sums of infinite series; use sequences and series to model real-life problems. Definition of Sequence: An ______________ sequence is a function whose _____________ is the set of positive integers. The function values 𝑎1 , 𝑎2 , 𝑎3 , … , 𝑎𝑛 , … are the terms of the sequence. If the __________ of the function consists of the first n positive integers only, the sequence is ____________. Copy Example 1 from page 642 in the space below. Copy Example 2 from page 643 in the space below. EX: Write the first five terms of each sequence. a.) 𝑎𝑛 = 2𝑛 + 1 b.) 𝑎𝑛 = 2 − (−1)𝑛 EX: Write the first five terms of 𝑎𝑛 = 2+(−1)𝑛 𝑛 Copy Example 3 from page 643 in the space below. EX: Write an expression for the nth term of each sequence. a.) 1,5,9,13,17 Notice that each term is ___ more than the previous. We need to think of a formula for each term in the sequence that has a factor of ___. In other words, _____ must be in the equation. b.) 1,3,5,7 Some sequences are defined ____________. In other words, you need the _____________ term to find the next term. Copy Example 4 from page 644 in the space below. EX: Find the first five terms of the sequence defined recursively by { 𝑎1 = 6 𝑎𝑘+1 = 𝑎𝑘 + 4 The ______________ Sequence is an example of a _____________ formula. 1, __, 2, ___, 5, 8, ___, 21, ____, ___, 89, . . . { 𝑎0 = 1 𝑎1 = ____ 𝑎𝑘 = ______________ Each term is found by adding the ________________ ____ terms. Factorial Notation: If n is a positive integer, n factorial is defined as n!=________________________. As a special case, ____ factorial is defined as 0!=__. Copy Example 5 from page 645 in the space below. EX: Write the first 5 terms of the sequence given by 𝑎𝑛 = 3𝑛 +1 𝑛! . Begin with n=0. The graph the terms on a set of coordinate axes. Now plot the points ______________________________________. Copy Example 6 from page 645 in the space below. EX: Evaluate each factorial expression. a.) 9! 3!8! b.) 4!4! 3!7! c.) (𝑛+1)! 𝑛! Summation Notation (__________ Notation): The sum of the first n terms of a _____________ is represented by ∑𝑛𝑖=1 𝑎𝑖 = ______________________________ where i is called the _____ of summation, n is the upper _______of summation, and 1 is the lower _____ of summation. Copy Example 7 from page 646 in the space below. EX: Find each sum. a.) ∑4𝑖=1(4𝑖 + 1) b.) ∑5𝑘=2(2 + 𝑘 3 ) c.) ∑6𝑖=1 2 𝑖! Series = sum of a _____________ Copy Example 8 from page 647 in the space below. 5 EX: For the series ∑∞ 𝑛=1 10𝑛 , find (a) the third partial sum and (b) the sum. 5 a.) ∑3𝑛=1 b.) We are trying to find the sum of an infinite sequence. What is happening to each successive term? Notice that the terms are approaching 0. Essentially, we are adding infinitesimally small values, or nothing at all. We can trick the calculator into solving this for us. 10𝑛 Copy Example 9 from page 648 in the space below. EX: The average cost to community hospitals per patient per stay from 1997 to 2004 is approximated by the model 𝑎𝑛 = 4.121𝑛3 − 89.65𝑛2 + 615.3𝑛 + 4907, 𝑛 = 0,1,2, … ,7, where 𝑎𝑛 is the cost in dollars and n represents the year with n=0 corresponding to 1997. Find the last three terms of this sequence, which represent the cost to community hospitals per patient per stay from 2002 to 2004. HW Pg.649 6-96 6ths, 125-128