9.2: Arithmetic Sequences & Partial Sums

As defined by the book, a sequence is arithmetic if the differences between consecutive terms are the same.

The common difference, or d, is the difference between each number in an arithmetic sequence.

An example of an arithmetic sequence would be something like this:

3, 5, 7, 9, 11, 13, 15, ...

This sequence has a common difference is 2.

There are a few equations that go along with arithmetic sequences and this section.                                                         First is the explicit formula:
                                  

         

Next is the recursive formula:

To find the nth term of an arithmetic sequence, use the formula:

Last is the formula to find the partial sum (the sum of the terms in a sequence):

I think that's about it for arithmetic sequences! And remember, never go in against a Sicilian, when death is on the line!

Jessica

           

9.1 Summation Notation

Summation notation is an easy way to represent the addition of the terms of a finite, or restricted, sequence. It uses the Greek letter Sigma.
 

In summation notation the first term of the sequence being added goes below the Sigma while the last term being added goes above.


Below is an example of summation notation:
 

In this case, the first term being added is the third term, because there is a 3 below the Sigma. The last will be the seventh, due to the 7 above the Sigma. The total will be the sum of the 3rd, 4th, 5th, 6th and 7th terms of the sequence.


Product Notation

While summation notation is the addition of the terms of a sequence, product notation is the product of the terms of a sequence multiplied together.  It uses the Greek letter Pi.



Like summation notation, the first term in of the sequence being multiplied goes below the Pi while the last term goes above.

 

In this case the 5th, 6th and 7th terms will be multiplied together to find the product of this portion of the sequence.



One Dozen Monkeys.
Which is always a let down, but for the sake of tradition must go on the blog.
http://www.youtube.com/watch?v=3nG1ckY7thw



Also you should know sun bears are cool




You already know Levitsky.

Beware the Wheel.

Don't be a book licker.

-Barton

9.1: Sequences and Summation Notation

Let's start with some basic definitions

• A Sequence is an ordered list of numbers
• Sequences can be written as an Explicit Formula or a Recursive Formula
• the Explicit Formula is used to determine any term a_n in a sequence.  Here is an example

Lets compare this to a function

 See that the two are very similar, but in a function, the domain is all real numbers.  In our sequence, the domain is natural numbers. 

•  the Recursive Formula is used to determine the next number in a sequence, or k=1, when given k

Notice that along with the initial equation, one is given one value in the sequence, here a₁, called the Initial Condition. 

Here are some handy patterns to know and be able to recognize

Lastly, Lets go over a short summary of Summation Notation using sigma.  when one wishes to add a certain number of terms in a sequence, it is more convenient to use Summation Notation than to write out each term.  Here is an example:

Where the number below the sigma is the starting term in the sequence, and the number on top is the ending term.  a_i is the explicit formula for the sequence.  

The terms in a sequence can also be multiplied using a similar notation

 

 This has been a general overview and review of Sequences and Summation Notation.

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