Introduction to Sequences - Problem 1
I want to talk about some notation that we use when we talk about sequences. What you see behind me is some common notation that we use. We have a, which refers to the term and then a little subscript which refers to the term number. So a sub 1, a subscript 1 is the first term, a sub 2 is the second term, a sub 3 is the third term and what we often write as a sub n is the nth term or what we call the general term.
Often times we want to find the general term because that will help us find any term in the sequence that we’re talking about.
For example, if I give you a sub 14 is 51 what this is saying is that we were to write out our sequence our 14th term is going to be 51. People sometimes have trouble with that they want to say this is the 51st term, no, the subscript is the term number and what that is equal to is what that term actually is.
Let’s look at a general term example. What we have here is a sub n s equal to 3n plus 1. What that’s saying is the term, whatever the term is, is dependent on what term number t is. Basically all we have to do is plugging the term number in order to figure out what that term is going to be. So a sub 1, this is our first term all we have to do is plug this 1 in for this n and see what comes out. So we plug 1 in, 3 times 1 is 3 plus 2 is 5. The first term in this case is 5.
A sub 2 is the second term. Let’s see what happens when we plug in 2. 3 times 2 is 6 plus 2 is 8. a sub 3, the third term. What happens when we plug in 3? 3 times 3 is 9 plus 2 is 11. And I ‘m actually going to change this example. What we, I had is a sub 4 but let’s jump a bit to a sub 11. What this is saying is what is the 11th term?
What’s cool about the general term is we don’t actually have to know everything before; we can actually jump to a specific term so a sub 11 is the 11th term all we have to do is plug in 11. 3 times 11 is 33 plus 2 is 35. If we can get our hands on the general term, it’s really easy for us to find any term in this sequence. We don’t need to know the first, second, third in order to find the 472 second. All we have to do is plug that in here and what comes out is that term.