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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