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**Unit 7: Sequences and Series**

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**Sequences A sequence is a list of #s in a particular order**

If the sequence of numbers does not end, then it is called an infinite sequence Each # in a sequence is called a term Ex. 3, 5, 7, 9…. a1=3 a2=5

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Arithmetic Sequences A sequence in which each term after the first term is found by adding a constant (called the common difference (d)), to the previous term Ex. Find the common difference (one term minus the previous term) 55, 49, 43, 37, 31, 25,19

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Arithmetic Sequences Formula for the general pattern for any arithmetic sequence:

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Arithmetic Sequences Write a equation for the nth term of the following sequence

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Arithmetic Sequences Write a equation for the nth term of the following sequence

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Arithmetic Sequences Find a15

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Arithmetic Sequences In the sequence below, which term has a value of 286?

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Arithmetic Sequences What is the value of the first term if the 9th and 10th terms are 4 and 2 consecutively?

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Arithmetic Sequences If the 3rd term of an arithmetic sequence is 8 and the 16th term is 47, find a1 and d

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Arithmetic Means Terms between any two non-successive terms of an arithmetic sequence Find 4 arithmetic means between 16 and 91

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Arithmetic Means Find 1 arithmetic mean between 50 and -120

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Results-4A Concerns: arithmetic means, notation, 2x problem, more practice

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Arithmetic Series A series is the indicated sum of the terms of a sequence If the seq. is 18, 22, 26, 30 The arith series is

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Arithmetic Series To find the sum of an arithmetic series,

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Arithmetic Series Find the sum of the first 100 positive integers.

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Arithmetic Series Find the sum of the first 20 even numbers beginning with 2.

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Arithmetic Series Find the sum of …+2

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Arithmetic Series Find the sum of …+97

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Geometric Sequences A geometric sequence is a sequence where each # in the seq. is multiplied by a constant (which is called the common ratio (r)) To find r, divide any term in the sequence by the previous term

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Geometric Sequences General Formula:

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Geometric Sequences Find the 11th term of the geo. Sequence listed below 64, -32, 16, -8,…a11

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Geometric Sequences Find the 6th term of the geo. sequence listed below 3, -15, 75, ..a6

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**Geometric Sequences Write an equation for the nth term**

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Geometric Sequences Find the 10th term of the sequence if a4=108 r=3

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Geometric Sequences Find the 7th term of the sequence if a3=96 r=2

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Geometric Means Geometric means are the missing terms between two non-successive terms in a geo. Sequence Find 3 geometric means between 2.25 and 576

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Geometric Means Find 5 geometric means between ½ and 1/1458

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Geometric Series A series that is associated with a geometric sequence

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Geometric Series Find the sum of the first 6 terms of the geometric series …

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Geometric Series What is r if the sum of the first 6 terms in a geo series is 11,718 and the first term is 3 Hint: solve by doing an intersection on the graph

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Geometric Series Find the first term of the series if the S8=39,360 and r=3

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Geometric Series Find the sum of the first 8 terms of 1+x+x2+x3+…

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**Geometric Series Find the a1 if Sn=165, an=48, r=-2/3 Hint: an=a1.rn-1**

So, r.an=a1.rn-1.r an.r=a1.rn (now substitute into the sum formula)

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Sigma Notation More concise (less time consuming) notation for writing out a series

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Sigma Notation

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Sigma Notation

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Sigma Notation

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**Write in sigma notation**

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**Write in sigma notation**

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**Write in sigma notation**

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**Write in sigma notation**

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**Write in sigma notation**

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**Infinite Geometric Series**

In an infinite series, Sn approaches some limit as n becomes very large. That limit is defined to be the sum of the series. If an infinite series has a sum, it is said to converge. A series converges (or has a sum) if and only if lrl < 1

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**Does the geom. series have a sum?**

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**Does the geom. series have a sum?**

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**Does the sum of each term approach some limit?**

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**To find the sum of an infinite series**

Make sure a limit exists first

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**An infinite series in sigma notation—find the sum**

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**An infinite series in sigma notation—find the sum**

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Pascal’s Triangle *First row is used for anything to the zero power **used for the coefficients of each term of the expanded binomial

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**Expand using Pascal’s Triangle**

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**Expand using Pascal’s Triangle**

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**Writing a repeating decimal as a fraction**

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**Writing a repeating decimal as a fraction**

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