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Section 9.2 Notes Arithmetic Sequences and Series

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  • Slide 1
  • Section 9.2 Notes Arithmetic Sequences and Series
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  • Definitions
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  • Sequence is a set of numbers arranged in a particular order. Term is one of the set of numbers in a sequence. Arithmetic Sequence is a sequence in which the difference between two consecutive numbers is constant. Common difference is this constant difference. Arithmetic Series is the indicated sum of the terms of an arithmetic sequence.
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  • # of Term (n)12345 Term (a n )710131619 Suppose the following the 1 st five terms of an arithmetic sequence are given
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  • What would be the next term? 22 How did you find the next term? Added 3 to 19. You used the recursive formula of an arithmetic sequence. The recursive formula for an arithmetic sequence is
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  • If these numbers were graphed as ordered pairs (n, a n ) what type of function would it be? Linear function What does the common difference represent in this function? Slope Write an equation for this arithmetic sequence in terms of a n. a n = 3(n 1) + 7 or a n = 3n + 4
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  • To find the value of any of the following in an arithmetic sequence: a 1 first term of the sequence n which term of the sequence d common difference of the sequence a n value of the n th term
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  • Use The explicit formula for an arithmetic sequence is a n = a 1 + (n 1)d or a n = dn + c, where c is the zero term
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  • C.To find one arithmetic mean between two numbers a and b use: D.To find the sum of an arithmetic sequence (arithmetic series) use:
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  • Or substitute a 1 + (n 1)d for a n
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  • E.Solve the following arithmetic sequence or series problems. 1.Find n for the sequence for which a n = 633, a 1 = 9, and d = 24 Steps: a. Use: a n = a 1 + (n 1)d
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  • b. Substitute the given values and solve for the unknown value. So 633 is the 27 th term of the given sequence.
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  • 2.Find the 58 th term of the sequence: 10, 4, 2, Steps: a. Find a 1, n, and d. a 1 = 10 n = 58 d = 6 b. Use: a n = a 1 + (n 1)d
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  • c. Substitute the given values and solve for the unknown value. So the 58 th term of the sequence is 332.
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  • An arithmetic mean of two numbers a and b is simply their average Numbers m 1, m 2, m 3, are called arithmetic means between a and b if a, m 1, m 2, m 3,, b forms an arithmetic sequence.
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  • 3.Form a sequence that has 5 arithmetic means between 11 and 19. Steps: a. The given sequence is 11, m 1, m 2, m 3, m 4, m 5, 19 b. a 1 = 11, n = 7, a n = 19
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  • c. Use: a n = a 1 + (n 1)d to solve for d. So the arithmetic sequence is: 11, 6, 1, 4, 9, 14, 19
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  • 4.Find n for a sequence for which a 1 = 5, d = 3, and S n = 440.
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  • n must be a positive integer so n = 16.
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  • 5.If a 3 = 7 and a 30 = 142, find a 20. Steps: a. Write two explicit equations to make a system of equations to solve for a 1 and d. b. In the first equation use a 30 = 142 and n = 30 to make the equation: 142 = a 1 + (30 1)d or 142 = a 1 + 29d. c. In the second equation use a 3 = 7 and n = 3 to make the equation: 7 = a 1 + (3 1)d or 7 = a 1 + 2d.
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  • d. Solve for d in the system of equations.
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  • e. Solve for a 1.
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  • f. Solve for a 20.
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  • 6.A lecture hall has 20 seats in the front row and two seats more in each following row than in the preceding one. If there are 15 rows, what is the seating capacity of the hall? a 1 = 20; d = 2; n = 15
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  • Summation Notation Summation notation is defined by where i is called the index, n is the upper limit, and 1 is the lower limit.
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  • Find the following sum. To do a sum in a graphing calculator depends on the calculator.