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Warm Up Evaluate. 1. ( - 1) 8 2. (11) 2 3. (–9) 3 4. (3) 4

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. 15. 19. Introduction to Sequences. Objectives: Find the nth term of a sequence.

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Warm Up Evaluate. 1. ( - 1) 8 2. (11) 2 3. (–9) 3 4. (3) 4

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  1. 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

  2. Introduction to Sequences Objectives: Find the nth term of a sequence. Write rules for sequences.

  3. 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}.

  4. 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.

  5. 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.

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

  7. Example 1: Finding Terms of a Sequence by Using a Recursive Formula Find the first 5 terms of the sequence with a1= –2 and an= 3an–1 + 2 for n ≥ 2. The first term is given, a1 = –2. Substitute a1 into rule to find a2. Continue using each term to find the next term. The first 5 terms are –2, –4, –10, –28, and –82.

  8. 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

  9. 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

  10. Find the first 5 terms of the sequence an=3n – 1. Example 2: Finding Terms of a Sequence by Using an Explicit Formula Make a table. Evaluate the sequence for n = 1 through n = 5. The first 5 terms are 2, 8, 26, 80, 242.

  11. Check It Out! Example 2a Find the first 5 terms of the sequence. an = n2 – 2n CheckUse a graphing calculator. Enter y1= 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 y1= 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.

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