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Warm Up

Look for a pattern and predict the next number or expression in the list. Warm Up. 62.5. 2. 3. 4. 5. 6. 22. 81. -12. 8. 3a+8b. Ch 13 sequences and series. Learn notation to define sequences, series, sums of series, and specific terms of either.

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Warm Up

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  1. Look for a pattern and predict the next number or expression in the list. Warm Up 62.5 2. 3. 4. 5. 6. 22 81 -12 8 3a+8b

  2. Ch 13 sequences and series • Learn notation to define sequences, series, sums of series, and specific terms of either. • Identify, find formulas, and find specific terms of sequences • Find sums, and formulas for sums, for finite series • Determine if an infinite series has a limit • If so, find the sum of the series

  3. notation • Continuous function vs sequence or series • Independent variable: xn • Dependent variable: y or f(x)tn • n is a subscript. Subscripts are counters. “t sub n”, means the value of term number n. • Ex. t5=12, means the value of term 5, is 12. • n is always a positive integer.

  4. 13.1 Arithmetic and Geometric Series Objective:To identify an arithmetic or geometric sequence and find a formula for its nth term. Chapter13 Sequences and Series

  5. A sequence is a set of numbers, called terms, arranged in some particular order. • An arithmetic sequence is a sequence with the difference between two consecutive terms constant.  The difference is called the common difference. • A geometric sequence is a sequence with the ratio between two consecutive terms constant.  This ratio is called the common ratio.

  6. Is each sequence arithmetic, geometric, or neither? What is the common difference or common ratio? 1)  3, 8, 13, 18, 23, . . .          2)  1, 2, 4, 8, 16, . . . 3)  24, 12, 6, 3, 3/2, 3/4, . . .              4)  55, 51, 47, 43, 39, 35, . . . 5)  2, 5, 10, 17, . . .       6)  1, 4, 9, 16, 25, 36, . . . 7) 3, 3, 3, 3, 3, …… • 1) Arithmetic, d = 5 • 2) Geometric, r = 2 • 3) Geometric, r = 1/2 • 4) Arithmetic, d = -4 • 5) Neither • 6) Neither • 7) Either, d = 0, r = 1

  7. Arithmetic Formula:tn  =  t1  +  (n - 1)d tnis the nth term, t1is the first term, n is the term number, and d is the common difference. • Geometric Formula:tn = t1. r(n - 1) • tnis the nth term, t1is the first term, n is the term number, and r is the common ratio.

  8. Find the first four terms and state whether the sequence is arithmetic, geometric, or neither. 1)  2) 3)  Find a formula for each sequence. 4) 5) 6)

  9. Find the first four terms and state whether the sequence is arithmetic, geometric, or neither. 1)  Arithmetic 1st Term: 2nd Term: 3rd Term: 4th Term: Common difference = 3 First four terms: 5, 8, 11, 14

  10. Find the first four terms and state whether the sequence is arithmetic, geometric, or neither. 1)  2) 3)  5, 8, 11, 14 Arithmetic d = 3 2, 5, 10, 17 Neither 6, 12, 24, 48 Geometric r = 2 Find a formula for each sequence. 4) 5) 6)

  11. Find a formula for each sequence. 4) Arithmetic t1 = 2,  the first number in the sequence d = 3, the common difference

  12. Find a formula for each sequence. 5) Geometric t1 = 4,the first number in the sequence r = 2, the common ratio

  13. Find a formula for each sequence. It's not geometric or arithmetic. 6) Think of the sequence as (20 +1), (200+1), (2000 + 1), (20000 + 1), . . . Then as this: [(2)(10) +1],[(2)(100) +1], [(2)(1000) +1], [(2)(10000) +1] Wait!  I see a pattern!   Powers of 10! tn = 2.10n + 1 Does this work? Try it and see!

  14. Find the indicated term of the arithmetic sequence with t1 = 5 and t7 = 29.   Find t53 8) Find the number of multiples of 9 between 30 and 901.

  15. Find the indicated term of the arithmetic sequence with t1 = 5 and t7 = 29.  Find t53

  16. 8) Find the number of multiples of 9 between 30 and 901. 36 What's the first multiple of 9 in the range? 900 What's the last multiple of 9 in the range? • Use the arithmetic formula. and solve for n

  17. Homework Page 476 #19-45

  18. Challenge 1. Findt7 for an arithmetic sequence wheret1 = 3x and d = -x. 2. Findt15 for an arithmetic sequence wheret3 = -4 + 5i  and  t6 = -13 + 11i

  19. 2. Find  t15 for an arithmetic sequence wheret3 = -4 + 5i  and  t6 = -13 + 11i Get a visual image of this problem Using the third term as the "first" term, find the common difference from these known terms. Now, from t3 to t15 is 13 terms.t15 = -4 + 5i + (13-1)(-3 +2i) = -4 + 5i -36 +24i     = -40 + 29i Challenge Answers 1. Find t7 for an arithmetic sequence wheret1 = 3x and d = -x. n = 7; t1 = 3x, d = -x

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