1. Unit 6: Modeling Mathematics 3 Ms. C. Taylor

2. Warm-Up

3. Simplifying Radicals

4. 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.

5. 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

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

7. 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

8. Let’s Try Some

10. Warm-Up

11. 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?

12. Graphing Radicals

13. Graphing Systems

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

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

16. 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

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

18. 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.

19. Given: Find: IDENTIFYSOLVE

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

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

22. Given: IDENTIFYSOLVE Find:

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

24. 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 ?

25. 71 + (-27) Each sum is the same. 50 Terms

27. Find the sum of the terms of this arithmetic series.

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

29. Alternate formula for the sum of an Arithmetic Series.

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

32. Warm-Up

33. 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

34. 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.

35. 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 )

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

45. 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

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

47. 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

48. 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 ✔

49. 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

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

51. 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

52. 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

53. 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 ✔

54. Warm-Up