Arithmetic Sequences

An arithmetic sequence is a type of sequence where the numbers increase (or decrease) by a common difference d.
To find a specific term in an arithmetic sequence, we could do two things. If we had a lot of time on our hands, we could just keep adding the common difference, but that's kind of frowned on by math teachers. The easier way to find a specific term is by using this formula:

In this formula, a(n) is the actual number you're trying to find. a(1) is the first term you're given. d is the common difference, and n is the number of the term you're on.
This might be a little tricky to understand with just words, so here's an example.

Sample sequence: 4, 7, 10, 13, 16...
We're going to find the 50th term in the sequence.
The difference d between each of these terms is a positive 3 added on to each. Since we're starting our sequence at 4, a(1)=4. The term number n we're trying to find is 50. Let's plug those in to our formula:

If we solve this, we find that a(50)=151.

Now let's say that we don't just want to find out what a particular term is - we want to sum up all the numbers up to that term. This is called a partial sum, and again we have two ways to find the value. We could manually add up all the numbers (again, frowned on by math teachers), or we could find it with this formula:

 Most of the terms in this formula represent the same things they represented in the formula for a particular term, but there are a couple of small changes. First of all, S(n) is the sum you're trying to find. Second, a(n) is not the term number you're trying to find but the term number at which you stop adding.