Sequences+and+Series


 * //ARITHMETIC SEQUENCES://**

When trying to determine the common difference of an arithmetic sequence or find the pattern, it is heplful to attempt to express each term in the sequence as an expression involving the same number. Find the common difference of the sequence:4,7,10,13...

Let //n=//the term number in the sequence. Let A(//n)=//the value of the //n//th term of the sequence.

A(1)=4(4+3)=7 __A(2)=7__(7+3)=10 A(3)=10 (10+3)=13 A(4)=13 (13+3)=16

The formula for the sequence is A(//n//)=4+(//n//-1)3 You could use the formula for the sequence to determind the next several terms simply by substituting a specific term in for //n//. For example: Term: 5th term: A(5)=4+(5-1)3 ___A(5)=16 100th term: A(100)=4+(100-1)3__ ___A(100)=301

Inductive reasoning:** is making conclusions based on pattens you observe. A conclusion you reach by inductive reasoning is a **conjecture.** A number pattern is also called a **sequence.** Each number in a sequnece is a **term**of the sequence. One kind of number sequence ia an arithmetic sequence. You form an **arithmetic sequence** by adding a fixed number to each previous term. This fixed number is the **common difference**. {Taken from: Algebra 1 textbook}
 * Key Terms:

(1) -4,5,14,23... What is the common difference? In each sequence you add 9 To get from -4 to 5 you add 9. To get from 5 to 14 you add an additional 9....
 * EXAMPLES:**

(2) -7,-3,1,5... What is the common difference? In each sequence you add 4 To get from -7 to -3 you add 4. To get from -3 to 5 you add an additional 4....

(3) 17,13,9,5.... What is the common difference? In each sequence you subtract 4 To get from 17 to 13 you subtract 4. To get from 13 to 9 you subtract an additional 4...

(4) -17,-22,-27,-32... What is the common difference? In each sequence you subtract 5 To get from -17 to -22 you subtract 5. To get from -22 to -27 you subtract an additional 5...

Suppose you are in a city and notice that the first three streets you pass are 10th Street, 11th Street, and 12th Street. You would probably conclude that the next street would be 13th Street. You would be basing your conclusion on inductive reasoning. {Taken from: Study Guide & Pratice Workbook}
 * Word problem:**

http://cs.jsu.edu/mcis/faculty/leathrum/Mathlets/seqser.html** [|**http://mathforum.org/library/drmath/sets/high_seq_series.html**]
 * Additional Links:


 * //GEOMETRIC SEQUENCES://**

Geometric sequences is just like arithmetic sequences except instead of ursing addition or subtraction you use multiplication or division. Find the next three terms of the seqenece 3,-9,27,-81...

Take the 2nd number divided by the first to find the common difference. -9/3= -3 The common ratio is -3 Note that each term in the given sequence is -3 times the previous term.

Let A(//n//)= the value of the //n//th term in the sequence. A(5)= -3*-81=243 A(6)= -3*243=-729 A(7)= -3*-729=2187 The next three terms in the sequence are 243, -729, 2187

Multiplying a term in the sequence by a fixed number to find the next term forms a **geometric sequence**. The fixed number is called the **common** **ratio**.
 * Key terms:**

(1) 3,12,48,192 What is the common ratio? In each sequence you multiply 4 To get from 3 to 12 you multiply 4. To get from 12 to 48 you multipy 4....
 * EXAMPLES:**

(2) 80,20,5,5/4 What is the common ratio? In each sequence you are dividing 4 that is the same as multiplying 1/4 In each sequence you multiply 1/4 To get from 80 to 20 you multiply 1/4. To get from 20 to 5 you multiply 1/4......