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Arithmetic and Geometric Sequences

I'm going to be honest straight off the bat, this topic varies in difficulty from person to person. A key part of sequences is patterns. For both geometric and arithmetic sequences, pattern recognition is by far the most useful tool to have. The only way I can think of displaying this is with an example.

Example 1: Suppose we have the following sequence:

0, 3, 6, 9, 12, 15, 18, 21, ........

What's the pattern for this sequence?

Well, the pattern here is that the next number is 3 plus whatever the previous number was. The first number for this sequence is called a(1), or sometimes it can be written as "a subscript 1." In a sequence, you are always given at least three terms of the sequence, if not more. Otherwise, if you only had the first value, the sequence could be anything.

Now, we know that a(1) = 0. Well, then the next number is obviously called a(2), and a(2) = 3. But, since we already know that pattern of the sequence, we can write down a general formula for this specific sequence. That is, for some nth number of the sequence, we can write down the (n+1)th number in terms of the nth number. This formula is:

a(n+1) = a(n) + 3

This is what is known as a recursive formula. Now, we have to put some conditions here. The number, n, cannot be negative (it technically can in some cases, but they are so very few and you probably won't see them, even if you did, you would treat them the same way as we are about to do right now). For all intents and purposes, let's just assume that n is always greater than or equal to 1.

But wait! We can do more than just create a recursive formula. We can create an expression for the nth term in terms of n! This allows us to calculate any member of the sequence by just plugging in its position in the sequence. 

Another way to state the pattern above is that each number in the sequence is just a multiple of 3. This allows us to write the sequence in a new way.

0*3, 1*3, 2*3, 3*3, 4*3 5*3, 6*3, 7*3, .....

This is the same sequence as above, just written a little differently. From this, we can see that number multiplied to three is the position, n, it is in, minus 1. Now we can write down the general formula for the nth term.

a(n) = (n-1)*3.

This is the formula for the nth term in the sequence. If you want, you can check for yourself by just plugging in n = 1, 2, 3, 4, 5, 6, ..... and seeing if you get the same sequence.

Now, example 1 was a bit of a simplistic example. The sequence was an arithmetic sequence. Some arithmetic sequences are more complex, involving multiplying a number and then subtracting another number from it. But one thing always remains the same, the next term of an arithmetic sequence is always the same distance away from the previous term. This is very important because it allows us to create a general formula for arithmetic sequences, which we will derive here and now.

Let's start with our first term, a(1). Normally, the first term is just referred to as "a" because that is typically the first term you are given. So now, let's create a(2) by adding some number, d, to a(1).

a(2) = a + d

remember: "a" is just another way of writing a(1).

The number added to a(1), d, is called the common difference. This is just the difference between two consecutive terms in an arithmetic sequence. Now, let's create a(3) by adding d to a(2).

a(3) = a(2) + d

But, we already have the formula for a(2) in terms of a(1) (= a). This means

a(3) = a + d + d = a + 2*d