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Unit 6: Modeling Mathematics 3 Ms. C. Taylor

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Warm-Up

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Simplifying Radicals

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Solving Radicals Step 1: Get rid of anything that might not be under the radical. Step 2: Square both sides to get rid of the radical. Step 3: Isolate the variable. Step 4: Solve Step 5: Plug answer back into equation and make sure that it works.

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What is a Radical Expression? A Radical Expression is an equation that has a variable in a radicand or has a variable with a rational exponent. yes no

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EXAMPLE – Solving a Radical Equation 2 () 2 Square both sides to get rid of the square root

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EXAMPLE 2 () 2 NO SOLUTION Since 16 doesn’t plug in as a solution. Let’s Double Check that this works Note: You will get Extraneous Solutions from time to time – always do a quick check

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Let’s Try Some

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Warm-Up

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Can graphing calculators help? SURE! 1.Input for Y1 2.Input x-2 for Y2 3.Graph 4.Find the points of intersection One Solution at (4, 2) To see if this is extraneous or not, plug the x value back into the equation. Does it work?

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Graphing Radicals

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Graphing Systems

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An Arithmetic Sequence is defined as a sequence in which there is a common difference between consecutive terms.

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Which of the following sequences are arithmetic? Identify the common difference. YES YES YES NO NO

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T h e c o m m o n d i f f e r e n c e i s a l w a y s t h e d i f f e r e n c e b e t w e e n a n y t e r m a n d t h e t e r m t h a t p r o c e e d s t h a t t e r m. C o m m o n D i f f e r e n c e = 5

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The general form of an ARITHMETIC sequence. First Term: Second Term: Third Term: Fourth Term: Fifth Term: nth Term:

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Formula for the nth term of an ARITHMETIC sequence. I f w e k n o w a n y t h r e e o f t h e s e w e o u g h t t o b e a b l e t o f i n d t h e f o u r t h.

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Given: Find: IDENTIFYSOLVE

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Given: Find: What term number is -169? IDENTIFYSOLVE

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Given: IDENTIFYSOLVE Find: What’s the real question?The Difference

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Given: IDENTIFYSOLVE Find:

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A r i t h m e t i c S e r i e s

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W r i t e t h e f i r s t t h r e e t e r m s a n d t h e l a s t t w o t e r m s o f t h e f o l l o w i n g a r i t h m e t i c s e r i e s. W h a t i s t h e s u m o f t h i s s e r i e s ?

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71 + (-27) Each sum is the same. 50 Terms

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Find the sum of the terms of this arithmetic series.

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Find the sum of the terms of this arithmetic series. What term is -5?

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Alternate formula for the sum of an Arithmetic Series.

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Find the sum of this series It is not convenient to find the last term.

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Warm-Up

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 33 Definition of Sequence An infinite sequence is a function whose domain is the set of positive integers. a 1, a 2, a 3, a 4,..., a n,... The first three terms of the sequence a n = 2n 2 are a 1 = 2(1) 2 = 2 a 2 = 2(2) 2 = 8 a 3 = 2(3) 2 = 18. finite sequence terms

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 34 Definition of Geometric Sequence A sequence is geometric if the ratios of consecutive terms are the same. 2, 8, 32, 128, 512,... geometric sequence The common ratio, r, is 4.

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 35 The nth Term of a Geometric Sequence The nth term of a geometric sequence has the form a n = a 1 r n - 1 where r is the common ratio of consecutive terms of the sequence. 15, 75, 375, 1875,... a 1 = 15 The nth term is 15(5 n-1 ). a 2 = 15(5) a 3 = 15(5 2 ) a 4 = 15(5 3 )

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 36 Example: Finding the nth Term Example: Find the 9th term of the geometric sequence 7, 21, 63,... a 1 = 7 The 9th term is 45,927. a n = a 1 r n – 1 = 7(3) n – 1 a 9 = 7(3) 9 – 1 = 7(3) 8 = 7(6561) = 45,927

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 37 Definition of Summation Notation The sum of the first n terms of a sequence is represented by summation notation. index of summation upper limit of summation lower limit of summation

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 38 The Sum of a Finite Geometric Sequence The sum of a finite geometric sequence is given by 5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ? n = 8 a 1 = 5

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 39 Definition of Geometric Series The sum of the terms of an infinite geometric sequence is called a geometric series. a 1 + a 1 r + a 1 r 2 + a 1 r 3 +... + a 1 r n-1 +... If |r| < 1, then the infinite geometric series has the sum

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 40 Example: Sum of Infinite Geometric Series Example: Find the sum of The sum of the series is

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 41 Graphing Utility: Terms and Sum of a Sequence Graphing Utility: Find the first 5 terms of the geometric sequence a n = 2(1.3) n. List Menu: variable Graphing Utility: Find the sum List Menu: beginning value end value variable upper limit lower limit

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Introduction Geometric sequences are exponential functions that have a domain of consecutive positive integers. Geometric sequences can be represented by formulas, either explicit or recursive, and those formulas can be used to find a certain term of the sequence or the number of a certain value in the sequence. 42 3.8.2: Geometric Sequences

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Guided Practice Example 1 Find the constant ratio, write the explicit formula, and find the seventh term for the following geometric sequence. 3, 1.5, 0.75, 0.375, … 43 3.8.2: Geometric Sequences

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Guided Practice: Example 1, continued 1.Find the constant ratio by dividing two successive terms. 1.5 ÷ 3 = 0.5 44 3.8.2: Geometric Sequences

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Guided Practice: Example 1, continued 2.Confirm that the ratio is the same between all of the terms. 0.75 ÷ 1.5 = 0.5 and 0.375 ÷ 0.75 = 0.5 45 3.8.2: Geometric Sequences

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Guided Practice: Example 1, continued 3.Identify the first term (a 1 ). a 1 = 3 46 3.8.2: Geometric Sequences

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Guided Practice: Example 1, continued 4.Write the explicit formula. a n = a 1 r n – 1 Explicit formula for any given geometric sequence a n = (3)(0.5) n – 1 Substitute values for a 1 and n. 47 3.8.2: Geometric Sequences

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Guided Practice: Example 1, continued 5.To find the seventh term, substitute 7 for n. a 7 = (3)(0.5) 7 – 1 a 7 = (3)(0.5) 6 Simplify. a 7 = 0.046875Multiply. The seventh term in the sequence is 0.046875. 48 3.8.2: Geometric Sequences ✔

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Guided Practice Example 3 A geometric sequence is defined recursively by, with a 1 = 729. Find the first five terms of the sequence, write an explicit formula to represent the sequence, and find the eighth term. 49 3.8.2: Geometric Sequences

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Guided Practice: Example 3, continued 1.Using the recursive formula: 50 3.8.2: Geometric Sequences

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Guided Practice: Example 3, continued The first five terms of the sequence are 729, –243, 81, –27, and 9. 51 3.8.2: Geometric Sequences

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Guided Practice: Example 3, continued 2.The first term is a 1 = 729 and the constant ratio is, so the explicit formula is. 52 3.8.2: Geometric Sequences

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Guided Practice: Example 3, continued 3.Substitute 8 in for n and evaluate. The eighth term in the sequence is. 53 3.8.2: Geometric Sequences ✔

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Warm-Up

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