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Splash Screen. Five-Minute Check (over Chapter 9) Then/Now New Vocabulary Example 1: Find Terms of Sequences Example 2: Recursively Defined Sequences Example 3: Real-World Example: Fibonacci Sequence Example 4: Convergent and Divergent Sequences Example 5: The n th Partial Sum

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  1. Splash Screen

  2. Five-Minute Check (over Chapter 9) Then/Now New Vocabulary Example 1: Find Terms of Sequences Example 2: Recursively Defined Sequences Example 3: Real-World Example: Fibonacci Sequence Example 4: Convergent and Divergent Sequences Example 5: The nth Partial Sum Key Concept: Sigma Notation Example 6: Sums in Sigma Notation Lesson Menu

  3. Find three different pairs of polar coordinatesthat name (–3, ) if –2π ≤ θ ≤ 2π. A. B. C. D. 5–Minute Check 1

  4. A.symmetric with respect to the line ; |r | = 4 when ; zeros for r when  =  ; B.symmetric with respect to the line ; |r | = 4 when ; zeros for r when ; C.symmetric with respect to the polar axis; |r | = 4 when = 0; zeros for r when = ; D.symmetric with respect to the polar axis; |r | = 4 when  = 0; zeros for r when = ; Determine the symmetry, zeros, and maximum r-values of r = 2 – 2 sin θ. Then use the information to graph the function. 5–Minute Check 2

  5. Write r = in rectangular form. A.x2 = 4(y + 1) B.y2 = 4(x + 1) C.x2 = 4(y – 1) D.(x + 1)= y2 5–Minute Check 3

  6. A. B. C. D. Which of the following represents 4 − 4i inpolar form? 5–Minute Check 4

  7. You used functions to generate ordered pairs and used graphs to analyze end behavior. (Lesson 1-1 and 1-3) • Investigate several different types of sequences. • Use sigma notation to represent and calculate sums of series. Then/Now

  8. sequence • term • finite sequence • infinite sequence • recursive sequence • explicit sequence • Fibonacci sequence • converge • diverge • series • finite series • nth partial sum • infinite series • sigma notation Vocabulary

  9. Find Terms of Sequences A. Find the next four terms of the sequence 3, –1, –5, –9, . . .. The nth term of this sequence is not given. One possible pattern is that each term is 4 less than the previous term. Therefore, a sample answer for the next four terms is 13, 17, 21, and 25. Answer:–13, –17, –21, –25 Example 1

  10. Find Terms of Sequences B. Find the next four terms of the sequence 18, 15, 10, 3, …. The nth term of this sequence is not given. If we subtract each term from the term that follows, we start to see a possible pattern. a2a1 = 15 – 18 or –3 a3a2 = 10 – 15 = –5 a4 a3 = 3 – 10 = –7 It appears that each term is generated by subtracting the next successive odd number. Therefore, a sample answer for the next four terms is 6, –17, –30, and –45. Answer:–6, –17, –30, –45 Example 1

  11. Find Terms of Sequences C. Find the first four terms of the sequence given by an = n3 + 1. Use the explicit formula to find an for n = 1, 2, 3, and 4. a1 = 13 + 1 or 2 n = 1 a3 = 33 + 1 or 28 n = 3 a2 = 23 + 1 or 9 n = 2 a4 = 43 + 1 or 65 n = 4 The first four terms in the sequence are 2, 9, 28, and 65. Answer:2, 9, 28, 65 Example 1

  12. A. 1, B. 1, C. D. 1, –3, –9, –27 Find the next four terms of the sequence 81, 27, 9, 3, …. Example 1

  13. Recursively Defined Sequences Find the fourth term of the recursively defined sequence a1 = 3, an = 3an – 1n + 2, where n 2. Since the sequence is defined recursively, all the terms before the fourth term must be found first. Use the given first term, a1 = 3, and the recursive formula for an. a2 = 3a2 – 1 – 2 + 2 n = 2 = 3a1 Simplify. = 3(3) or 9 a1 = 3 a3 = 3a3 – 1 – 3 + 2 n = 3 = 3a2 – 1 or 26 a2 = 9 Example 2

  14. Recursively Defined Sequences a4 = 3a4 – 1 – 4 + 2 n = 4 = 3a3 – 2 or 76 a4 = 76 The fourth term is 76. Answer:76 Example 2

  15. Find the fifth term of the recursively defined sequence a1 = 28, an = an – 1 + 4, where n 2. A. B. C. D. Example 2

  16. Fibonacci Sequence NATURE Suppose that a stem has to grow for two months before it is strong enough to support branches. At the end of the second month, it sprouts a new branch and continues to sprout one new branch each month. The new branches grow similarly. How many branches will a plant have after 18 months if no branches are removed? During the first two months, there will only be one branch, the stem. At the end of the second month, the stem will produce a new branch, making the total for the third month two branches. The new branch will grow and develop two months before producing a new branch of its own, but the original branch will now produce a new branch each month. Example 3

  17. Fibonacci Sequence The table below shows the pattern. Each term is the sum of the previous two terms. This pattern can be written as the recursive formula a1 = 1, a2 = 1, an= an – 2 + an– 1 , where n ≥ 3. Therefore, to find the number of branches after 18 months, continue the table until you find the number of branches after 16 months and 17 months. Example 3

  18. Fibonacci Sequence The number of branches are shown in the table below. If no branches are removed, a plant will have 2584 branches after 18 months. Answer:2584 branches Example 3

  19. NATURE Suppose that a stem has to grow for two months before it is strong enough to support branches. At the end of the second month, it sprouts a new branch and continues to sprout one new branch each month. The new branches grow similarly. How many branches will a plant have after 20 months if no branches are removed? A. 1597 B. 2584 C. 4181 D. 6765 Example 3

  20. A. Determine if the sequence is convergent or divergent. Convergent and Divergent Sequences The first eight terms of this sequence are given or approximated below. a1 = 1000 a2 = 250 a3 = 62.5 a4 = 15.625 a5 = 3.90625 a6 = 0.97656 a7 = 0.24414 a8 = 0.06104 From the graph at the right, you can see that an approaches 0 as n increases. This sequence has a limit and is therefore convergent. Example 4

  21. Convergent and Divergent Sequences Answer:convergent Example 4

  22. B. Determine if the sequence is convergent or divergent. Convergent and Divergent Sequences The first eight terms of this sequence are –1, 2, –4, 8, –16, 32, –64, and 128. From the graph below, you can see that an does not approach a finite number. Therefore, this sequence is divergent. Example 4

  23. Convergent and Divergent Sequences Answer:divergent Example 4

  24. C. Determine if the sequence is convergent or divergent. Convergent and Divergent Sequences The first eight terms of this sequence are given or approximated below. a1 = 0.1 a2 = 0.00667 a3 = 0.0006 a4 =0.000057 a5 = 0.0000056 a6 = 0.00000055 a7 = 0.000000054 a8 = 0.0000000053 Example 4

  25. Convergent and Divergent Sequences From the graph below, you can see that an approaches 0 as n increases. This sequence has a limit and is therefore convergent. Answer:convergent Example 4

  26. Determine whether the sequence is convergent or divergent. A. convergent B. divergent Example 4

  27. The nth Partial Sum A. Find the fifth partial sum of an = n2– 3. Find the first five terms. a1 = (1)2 – 3 or –2 n = 1 a2= (2)2 – 3 or 1 n = 2 a3 = (3)2 – 3 or 6 n = 3 a4 = (4)2 – 3 or 13 n = 4 a5 = (5)2 – 3 or 22 n = 5 The 5th partial sum is S5 = –2 + 1 + 6 + 13 + 22 or 40. Answer:40 Example 5

  28. B. Find S4 of . a1 = or 3 n = 1 a2= or 1.5 n = 2 a3 = or 0.75 n = 3 a4 = or 0.375 n = 4 The nth Partial Sum Find the first four terms. Example 5

  29. The 4th partial sum is S4 = 3 + 1.5 + 0.75 + 0.375 = 5.625 or . Answer:5.625 or The nth Partial Sum Example 5

  30. Find the fifth partial sum of an = (–3)n – 2. A. –246 B. –193 C. 365 D. 534 Example 5

  31. Key Concept 6

  32. A. Find . = [(1)2 – 1] + [(2)2 – 2] + [(3)2 – 3] + [(4)2 – 4] Sums in Sigma Notation = 0 + 2 + 6 + 12 = 20 Answer:20 Example 6

  33. B. Find . = 15.85 or Answer:15.85 or Sums in Sigma Notation = 4.5 + 4 + 3.75 + 3.6 Example 6

  34. C. Find . = 0.33333… or Answer: Sums in Sigma Notation = 0.3+ 0.03 + 0.003 + 0.0003 + 0.00003 + … Example 6

  35. Find . A. 92 B. 123 C. 215 D. 429 Example 6

  36. End of the Lesson

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