Series

A ** sequence ** is a list of numbers that has a certain order. For example, 16,7,888,-2,0,72 is a sequence, since 16 is the first number in the list, 7 is the second, etc. Sequences can have a certain "rule" by which terms progress, but they can also be completely random. The sequence of natural numbers, 1, 2 , 3 , 4 , 5  , ...   and the sequence of odd natural numbers, 1, 3 , 5 , 7 , 9  , ...


 * Arithmetic sequences ** are sequences that start with any number a, and in which every [[image:http://texhub.com/b/bl4ge3RofQ== caption="n^ {th}"]]term can be written as [[image:http://texhub.com/b/YV9uPWErKG4tMSlk caption="a_n=a+(n-1)d"]], where d is any number. An example of such a sequence would be 5, 12, 19, 26, 33, where [[image:http://texhub.com/b/YT01 caption="a=5"]]and [[image:http://texhub.com/b/ZD03 caption="d=7"]]. This is an increasing arithmetic sequence, as the terms are increasing. Decreasing arithmetic sequences have [[image:http://texhub.com/b/ZDww caption="d<0"]].


 * Geometric sequences ** also start with any number a (though usually a is nonzero here), but this time we are not adding an extra d value each time- we multiply a by a factor of r. Thus, the [[image:http://texhub.com/b/biBeIHt0aH0= caption="n ^ {th}"]]term is [[image:http://texhub.com/b/YV9uPWFyIF4ge24tMX0= caption="a_n=ar ^ {n-1}"]]. Geometric sequences can either be monotonic, when r is positive and the terms are moving in one direction, or alternating, where [[image:http://texhub.com/b/cjww caption="r<0"]]and the terms alternate between positive and negative values, depending on n.

A ** series ** is a sequence of numbers that represent partial sums for another sequence. For example, if my sequence is 1,2,3,4 then my series would be 1,1+2,1+2+3,..., or 1,3,6,10. With arithmetic and geometric series, we can use a formula to calculate any term of the series. These shortcuts are useful since they save you from having to write out the entire sequence and add all of its terms up. The formula for an arithmetic series is: The formula for a geometric series is: These formulas are worth memorizing.

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