Sequences Part I

In one of my past math posts (12/21), I talked about permutations and combinations. Today, we will be stepping off of that and getting into a little bit of series/sequences.

First of all, we need to know what sequences are. A sequence is a list of objects presented in a particular order. The objects in a sequence are called the terms of the sequence. For example, 2,4,6… is a sequence of positive even numbers. The first term here is 2, the second term is 4, and so on. Pretty simple concept. Now, another simple thing to know is that the index of a term is the position of a term. In our previous example, 4’s index is 2 because it is the second term. Mathematicians notate this as x2=4, with the index being the subscript.

Sometimes, there is a relationship between the index and the term. For instance, notice that in even numbers, the term is two times its index. For instance 4 is two times its index-2, and 6 is two times its index-3. We can notate this as { xi=2i}. Let’s try out this question:

What sequence is generated by {xi=i^2}?

Well, lets start solving for the first term, which would equal x1. Meaning i=1. So we plug in 1 into i^2, which would be 1^2=1. So 1 is our first term. For our second term, we also simply go through the same process, plugging 2 into i^2, which would equal 4. For the third, we plug in 3, equaling 9. For 4, we plug it in equaling 16. So the sequence goes like this: 1,4,9,16….

Try out this question for yourself. What formula will generate the sequence 3,7,11, 15…? Do the formula in terms of  xi. If you can’t answer and want to know how, contact me.

So far, all the sequences we have considered are infinite sequences, meaning they go on forever. However, not all sequences are like this. Some do end, or are finite. For instance, let’s say we shorten the infinite sequence of positive even numbers to start at 2 and stop at 10. We would notate this the same way as we did with the infinite sequence of even numbers, but with a few exceptions. After the end bracket, we would add a subscript (starting with i=) to denote the starting value as the index. In this case, the subscript would be i=1, because 2 is the first term. We would also add a superscript, denoting the ending value of the index, which in this case is 5 because 10 is the 5th term.

In this type of subject, there are two kind of formulas: recursive and direct formulas. Recursive formulas are formulas for a sequence that declare the starting value for that sequence and how a subsequent term is made from the previous term. For instance, take the sequence 3,7,11,15,19… We first declare the starting value like this: x1=3. Then we add a semicolon. We know that in order to get from one term to the next ,we add four. So let xbe the term we want to find. The previous term would be xi-1 and for the term to move onto xi, we add four so the recursive formula is xi= xi-1+4. Note that in order to use this, we have to know the previous term xi-1.

So what are direct formulas? It is any formula where xis expressed not in terms of xi-1, for instance like {xi=i^2}. It is a formula that you state directly and can be used without knowing the previous term.

For tomorrow, I will introduce the two types of sequences.

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