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12-1. Introduction to Sequences. Warm Up. Lesson Presentation. Lesson Quiz. Holt Algebra 2. Warm Up Evaluate. 1. ( - 1) 8 2. (11) 2 3. (–9) 3 4. (3) 4 Evaluate each expression for x = 4. 5. 2 x + 1 6. 0.5 x + 1.5 7. x 2 - 1 8. 2 x + 3. 1. 81. 121. –729. 9. 3.5.

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12-1

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  1. 12-1 Introduction to Sequences Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

  2. Warm Up Evaluate. 1. (-1)8 2. (11)23. (–9)34. (3)4 Evaluate each expression for x = 4. 5. 2x + 1 6. 0.5x + 1.5 7. x2 - 1 8. 2x + 3 1 81 121 –729 9 3.5 15 19

  3. Objectives Find the nth term of a sequence. Write rules for sequences.

  4. Vocabulary sequence term of a sequence infinite sequence finite sequence recursive formula explicit formula iteration

  5. In 1202, Italian mathematician Leonardo Fibonacci described how fast rabbits breed under ideal circumstances. Fibonacci noted the number of pairs of rabbits each month and formed a famous pattern called the Fibonacci sequence. A sequenceis an ordered set of numbers. Each number in the sequence is a term of the sequence. A sequence may be an infinite sequencethat continues without end, such as the natural numbers, or a finite sequencethat has a limited number of terms, such as {1, 2, 3, 4}.

  6. You can think of a sequence as a function with sequential natural numbers as the domain and the terms of the sequence as the range. Values in the domain are called term numbers and are represented by n. Instead of function notation, such as a(n), sequence values are written by using subscripts. The first term is a1, the second term is a2, and the nth term is an. Because a sequence is a function, each number n has only one term value associated with it, an.

  7. In the Fibonacci sequence, the first two terms are 1 and each term after that is the sum of the two terms before it. This can be expressed by using the rule a1 = 1, a2 = 1, and an= an – 2 + an – 1, where n ≥ 3. This is a recursive formula. A recursive formulais a rule in which one or more previous terms are used to generate the next term.

  8. Reading Math anis read “a sub n.”

  9. Check It Out! Example 1a Find the first 5 terms of the sequence. a1 = –5, an = an –1– 8 a an –1 – 8 an 1 Given –5 2 –5 – 8 –13 3 –13 – 8 –21 4 –21 – 8 –29 5 –29 – 8 –37

  10. Check It Out! Example 1b Find the first 5 terms of the sequence. a1 = 2, an= –3an–1 a –3an –1 an 1 Given 2 2 –3(2) –6 3 –3(–6) 18 4 –3(18) –54 5 –3(–54) 162

  11. Check It Out! Example 2a Find the first 5 terms of the sequence. an = n2 – 2n CheckUse a graphing calculator. Enter y = x2 – 2x. n n2 – 2n an 1 1 – 2(1) –1 2 4 – 2(2) 0 3 9 – 2(3) 3 4 16 – 2(4) 8 5 25 – 2(5) 15

  12. Check It Out! Example 2b Find the first 5 terms of the sequence. an = 3n – 5 CheckUse a graphing calculator. Enter y = 3x – 5. n 3n – 5 an 1 3(1) – 5 –2 2 3(2) – 5 1 3 3(3) – 5 4 4 3(4) – 5 7 5 3(5) – 5 10

  13. In some sequences, you can find the value of a term when you do not know its preceding term. An explicit formuladefines the nth term of a sequence as a function of n.

  14. Remember! Linear patterns have constant first differences. Quadratic patterns have constant second differences. Exponential patterns have constant ratios. (Lesson 9-6)

  15. 1st differences –2 –2 –2 –2 Check It Out! Example 3a Write a possible explicit rule for the nth term of the sequence. 7, 5, 3, 1, –1, … Examine the differences and ratios. The first differences are constant, so the sequence is linear.

  16. 1st differences –2 –2 –2 –2 Check It Out! Example 3a Continued The first term is 7, and each term is 2 less than the previous. A pattern is 9 – 2n. One explicit rule is an = 9 – 2n.

  17. A pattern is One explicit rule is an = Check It Out! Example 3b Write a possible explicit rule for the nth term of the sequence.

  18. Check It Out! Example 4 An ultra-low-flush toilet uses 1.6 gallons every time it is flushed. Graph the sequence of total water used after n flushes, and describe its pattern. How many gallons have been used after 10 flushes? The graph shows the points lie on a line with a positive slope; 16 gallons have been used after 10 flushes.

  19. Recall that a fractal is an image made by repeating a pattern (Lesson 5-5). Each step in this repeated process is an iteration, the repetitive application of the same rule.

  20. Check It Out! Example 5 The Cantor set is a fractal formed by repeatedly removing the middle third of a line segment as shown. Find the number of segments in the next 2 iterations. By removing the middle third of a line segment, each iteration results in a line segment splitting into two line segments. So the number of line segments doubles with each iteration. The number of line segments can be represented by the sequence an = 2n – 1. The next 2 iterations, the 4th and 5th terms are a4 = 24 - 1 = 8 and a5 = 25– 1 = 16.

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