3rd Hour Honors Algebra 2 B (Spring 2012): 10.2: Arithmetic Sequences

An Arithmetic Sequence is a sequence that has a common difference (d) between each term.

$a_{k+1}-a_{k}=a_{j+1}-a_{j}$

For any numbers/terms j and k. Arithmetic Sequences are linear, they increase by the same amount each term. Finding the Common Difference (d):

$d=a_{k+1}-a_{k}$

Don't be afraid to do it in your head... Examples: -3, 2, 7, 12, ... Common Difference = +5 17, 10, 3, -4, ... Common Difference = -7 The common difference is constant!!! This means that it always stays the same no matter what two consecutive terms next to each other that you measure the difference of. Another way to look at it is this:

$a_{k+1}=a_{k}+d$

You can also use that to prove that the sequeuence is infant arithmetic. Finding the nth term of an arithmetic sequence:

$a_{n}=a_{1}+(n+1)d$

Example: Find the 63rd term of a sequence when d=4 and the first term is 12.

$a_{63}=12+(62)4$

$a_{63}=12+248$

Therefore the 63rd term would be 260. This works because you are taking the starting point plus the common difference time the term you need minus one, so that you don't count the first term as zero. Finding the Sum:

$S_{n}=\frac{n}{2}(a_{1}+a_{n})$

n is the number of the terms that you are finding the sum of. But first you must find a sub n with that formula. The Partial Sum can also be defined as:

$S_{n}=a_{1}+a_{2}+a_{3}+...+a_{n}$