Sequences+&+Series

__ Instructional Video Links: __
[] This video shows ways to solve geometric and arithmetic sequences.(Abbey)

[] This song talks about geometric series and partial sums. It also talks about limits. (Abbey)

[] This video talks about finding the sum of an arithmetic sequence. It shows two formulas for how to find it, one we've used and one he uses, but they both work either way. It also shows you how to find partial sums using the explicit formula for certain terms needed. (Mandi)

[] This video talks about finding the sum of a geometric sequence. (Mandi)

[] This video talks about summation notation. It tells what sigma, n, and i = is. (Mandi) (Abbey)

__ Jobs and Occupations that use Sequences and __ __Series:__ ** (Nicholas) **
Design insurance plans that will help their company make a profit. They make insurance plans based on statistics and social trends to decide what to charge and predict what the company will have to pay for customer claims.
 * Actuaries**:

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Use special computer controlled machines to shape things like machine parts, car parts, and compressors. They must constantly monitor computer readouts to make sure that the parts are being made properly.
 * Computer Control Programmers and Operators**:

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__ Helpful Links for Sequences and Series: __
[] This website is a great website which almost everything you would need to know for a sequence. This is an awesome website and I would recommend that this website be used for the basics of an Arithmetic or Geometric sequence. ( Elijah K.)

[] This website helps you solve infinite Geometric series. It provides the basic formula as well as the rule when you should actually solve for the infinite series. This website also provides an example problem which is worked out step by step to make the problem easy. ( Elijah K.)

[] This website helps the student to determine whether the given sequence is an Arithmetic or Geometric sequence.The website also provides many other links within the same website, that help you go more in depth with an Arithmetic or Geometric sequence. This is a very good website to review this unit's material over sequences. ( Elijah K.)

[] This elaborate website provides students with step-by-step videos, practice problems, and instructional steps on how to create or use a formula, to find a specific term number or the sum, for an Arithmetic or Geometric sequence. It will also show you the basics on how to determine whether a sequence is Arithmetic or Geometric. If the student needs extra help, or has any further questions or concerns, this website also provides a section under each step-by-step video where the student may ask any desired questions. The student will then receive helpful feedback from other students who have mastered the concept of the asked concerned question. Not only does this website concentrate on Arithmetic and Geometric sequences, it also elaborates upon other math concepts. I would recommend this website to any student who struggles in math and needs the extra help. (Stephanie M.)

[] This website elaborates upon Arithmetic and Geometric sequences fairly well. To find the sum of an Arithmetic or Geometric sequence is discussed upon, as well as how to determine whether a sequence has a common difference or a common ratio. This website also provides step-by-step examples on how to evaluate higher level problems (to find a specific term or sum), from Arithmetic sequences, by using logarithms. As for Geometric sequences, finite and infinite sequences are discussed and provided each with an example. It also talks upon and provides examples on how to find the Arithmetic or Geometric mean when only two numbers are given. (Stephanie M.)



** 1. Find a44 **
4,10, 16, 22, 28...

** 2. Find a64 **
10, 2, -6, -14...



** 3. Find the sum of the geometric sequence. **
-44 + -22 + -11 + -5.5 + ... -.69

i=1
a.) 7 b.) 11/6 c.) -16/11 d.) -11/16 e.) 1/6

** 4.) Write in summation notation. **
102 + 58 + 14 + -30 + -74 ... -250

**5.) (Nicholas)**
====** You go to a shop and you find a rare baseball card that is worth $500. The clerk at the store says that the value of the value of the card increases 3% every year. If you buy the card and wait seven years to sell it from the day you buy it, how much will it be worth? **====

**Convert the following recursive formula to the explicit formula:**
a^1=7 a^n=a^n-1+7

__ Arithmetic Summary __ (Elijah K.)
An arithmetic sequence is a list of numbers in which the difference between consecutive are constant. An Arithmetic sequence can start with any number, but must always have the same common difference (d).

Formulas:
__Recursive:__ //These formulas can be used to find any term within an Arithmetic sequence, as long as there is a constant common difference.// {A1 = start {An= An-1 + d

__Explicit:__ An= A1 + d (n-1)

Summation Formula :
Sn= n ( A1 + An ) / 2 T // his formula allows you to find the sum of any arithmetic series. //


 * n = number of terms in the sequence
 * A1 = the first term of the sequence
 * An = the number from the desired term
 * d = common difference between each term

Examples:
Find the 10 term of the following terms: 2,7,12,17 The common difference between these terms is 5. You would then either setup an explicit or recursive formula. A10=A1 + d(n-1). Now that we have an explicit formula plug in your given information. A10=2 + 5(10-1) which would equal 2+5(9) which gives you the term 47. So we can conclude that the 10th term of this sequence is 47.

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**__Arithmetic Sequences: A Formula for the ' n - th ' Term__** By: patrickJMT
(Stephanie M.)

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__ Geometric Summary __** (Stephanie M.) **
A Geometric sequence is a list of numbers where each term is a multiple of the previous one (each term after the first is found by multiplying the previous one). The factor being multiplied is called the common ratio (a fixed non-zero number) known as "r". When multiplying to get to the next term in a specific sequence, the common ratio will be the same each time you multiply.

**Formulas:**
//These formulas can be used to find any term in a Geometric sequence.//

__Recursive__- {a1= start {an= (r)(an-1)

__Explicit__- an=a1(r)^n-1

**Sum Formula:**
//These formulas can be used to determine the sum of a Geometric sequence.//

__Finite__- sn= a1(1-r^n)/1-r

__Infinite-__ (Convergent |r|<1)- sn= a1/1-r (Divergent |r|>1)- "Undefined"

Meaning of Symbols:


 * n = The number of terms in the series."When in doubt, count 'em out!"
 * r = the common ratio
 * a1 = the first term of the series
 * an = the last term of the series.

**Examples:**
a.) In the sequence 400, 200, 100, 50...., what is the 8th term?

First, we determine whether this sequence is an Arithmetic of Geometric sequence. We find that this sequence is Geometric, so then we find the common ratio which, in this case, is 0.5. Second, you determine whether you want to use the explicit formula or the recursive formula for Geometric. I am going to be using the explicit formula because I find it easier and faster to get to my final answer. When we set up the explicit formula for this Geometric series, we simply plug in the information that we found and that was given to us to get the following formula a8 = 400(0.5)^8-1. The formula, 400(0.5)^8-1, simplifies out to 400(0.5)^7 which then simplifies even more to the number 3.125. Therefore, concluding that the eighth term of this Geometric sequence is 3.125.

b.) Given the following information, create an explicit formula: a4 = 0.04, r = 0.2


 * 1) 0.04 = a1(0.20)^4-1 Plug in the given information (Hint: You have to find a1)
 * 2) 0.04 = a1(0.2)^3 Simplify
 * 3) 0.04 = a1(0.008) Divide to get a1 by itself
 * 4) a1 = 5 Plug in a1 and "r" to get the correct explicit formula
 * 5) an = 5(0.2)^n-1

** __Geometric Sequences: A Formula for the' n - th ' Term__ By: patrickJMT **
(Stephanie M.)

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**__Sum of an Infinite Geometric Series, Ex 2__** By: patrickJMT
(Stephanie M.)

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**Annotated Bibliography ** Timothy Kurdt and Jenna Labbie ||
 * ** Mathematics 1001: Absolutely Everything That Matters About Mathematics in 1001 Bite-Sized Explanations ** By Dr. Richard Elwes