Unit 10: Sequences & Series By: Saranya Nistala

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

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

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

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

Arithmetic and geometric sequence formulas

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 : ∞ i=1 6(0.6) i = = n

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 (-) 3434 n = =

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

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

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

Connections to other topics  Exponential growth and decay models- connected to graphing geometric sequences  Statistics- connected because both use summation notation