Arithmatic Sequences and Partial Sums

Definition of an Arithmetic Sequence:

A sequence is arithmetic, if the differences between consecutive terms are the same, so if you have  a1, a2, a3, a4, ...
Then,

a2 - a1 = a3 - a2 = a4 - a3 = ..... d

d is the common difference of the arithmetic sequence


An example of an arithmetic sequence would be 3, 6, 9, 12, 15...

since 6-3 = 3,   9-6=3,  12-9=3,  15-12=3 ... and so on.

In this case, d = 3; The common difference of this arithmetic sequence is 3.

Finding The nth Term of an Arithmetic Sequence:





where d is the common difference in the consecutive terms of the sequence and 


 An example of finding the nth term would be- in the sequence  3,8,13,18...  you need to simply find the common difference which would be 8-3=5, with the common difference, place it in front of n in your equation which is now . With this equation, plug in a number from the sequence lets try 3,
 ; solve for c; c=-2. so plug d and c back into the original equation of  and you get .

The Sum of a Finite Arithmetic Sequence:


The sum can be found with the equation where sn is the sum, and n is the number of terms.

An example of this equation would be- With the sum of 1+3+5+7+9, replace n with 5 (since there are five terms), a1 with 1, an with 9. The new equation is . Solve and you get , So the sum of these five terms = 25.