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Unit 10: Sequences & Series By: Saranya Nistala
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Unit Goal: I can find and analyze arithmetic and geometric sequences and series. Key Concepts: Define & use sequences and series Analyze arithmetic sequences and series Analyze Geometric sequences and series Find sums of infinite geometric series Use recursive rules with sequences and functions
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Define & use sequences and series Important Terms: Sequence- a function whose domain is a set of consecutive integers Terms- the values in the range of a sequence Series- the expression that results when the terms of a sequence are added together Summation notation( Ʃ )- the notation for a series that represents the sum of the terms Things we learned: Describe the pattern Write the next term Write a rule for the nth term of the sequence Write the series using summation notation Find the sum of a series Formulas for special series: Ʃ Ʃ Ʃ 1 = n n i=1 n i = ------- n(n+1) 2 i = -------------- n i=1 n(n+1)(2n+1) 6
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Analyze arithmetic series and sequences Important terms: Arithmetic sequence- a sequence in which the difference between consecutive terms is constant Common difference- the constant difference between terms of an arithmetic sequence denoted by d Arithmetic series- the expression formed by adding the terms of an arithmetic sequence denoted by S Things we learned: Tell if the sequence is arithmetic Given 1 term and common difference, write a rule for the nth term Given 2 terms, writhe a rule for the nth term Find the sum of an arithmetic series n
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Analyze geometric sequences and series Important terms: Geometric sequence- a sequence in which the ratio of any term to the previous term is constant Common ratio- the constant ratio between consecutive terms of a geometric sequence denoted by r Geometric series- the expression formed by adding the terms of a geometric sequence Things we learned: Tell if the sequence is geometric Given 1 term and common ratio, write a rule for the nth term Graph a geometric sequence Given 2 terms, write a rule for the nth term Find the sum of a geometric series
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Arithmetic and geometric sequence formulas
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Find sums of infinite geometric series Important terms: Partial sum- the sum S of the first n terms of an infinite series Things we learned: Find the sum of an infinite geometric series Write a recurring decimal as a fraction in lowest terms Formula for infinite geometric series: S = -------- a 1 1- r Example 1: Ʃ Solution : 6 6 15 1- 0.6 0.4 ∞ i=1 6(0.6) i-1 ------------ = ------ = n
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Real life situation… Example 2: A person is given one push on a swing. On the first swing a person travels a distance of 4 feet. On each successive swing, the person travels 75% of the distance of the previous swing. What is the total distance that the person swings? Solution: *First write the rule for the sequence 4 x 4 4 1- --- --- (-) 3434 n-1 ------------ = ----------- = 16 3434 1414
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Use recursive rules with sequences and functions Important terms: Explicit rule- a rule for a sequence that gives a as a function of the terms position number n Recursive rule- a rule for a sequence that gives the beginning term of terms of a sequence and then a recursive evaluation that tells how a is related to one or more preceding terms Iteration- the repeated composition of a function f with itself Things we learned: Write a recursive rule for the sequence Find the iterates of a function n n
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Practice problems Example 3: Write a recursive rule for the sequence 3,5,2,-3,-5……. Solution: a = a - a Example 4: Find the first three iterates x, x, x of the function f(x) = 2x – 5 for an initial value of x = 3. Solution: x = f(x ) x = 1, x = -3, x = -1 n n-1 n-2 2 3 1 1 0 1 2 3
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Common mistakes and struggles Applying the correct formula for a problem When writing a recurring decimal as a fraction in lowest terms, separate non-repeating and repeating parts Always use a when taking an even root When solving a story problem, write the rule for the sequence first, to avoid mistakes When finding the sum for an infinite geometric series, r must be less than 1
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Connections to other topics Exponential growth and decay models- connected to graphing geometric sequences Statistics- connected because both use summation notation
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