Introduction to Sequences - Problem 2
Finding the general term for a sequence. Whenever we’re trying to find the general term, what we’re looking for is a sub n and when we’re doing that what we have to do is somehow find dome patterns in the sequences that we’re looking at and figure out how to relate them to a general case. The cool thing about the general term is we should be able to find any term in the sequence without having to know the term before it.
My first example is 4, 8, 12, 16, so on and so forth. And what we’re doing is looking for patterns that relates to the sequence. And what we want to do is break, I always try to break this down into what they have in common. Looking at these what I see is they’re all multiples of 4. So I often rewrite it knowing that information.
So what we have is then 1 times 4, 2 times 4, 3 tims4, and 4 times 4 and then we’re going to have a bunch of other terms after that. This is our first term. This is our second, this is our third, this is our fourth which corresponds to what is being multiplied by 4. We have 1 times 4, the first term times 4, 2 times 4, the second term is being multiplied by 4 as well. So what we really end up with is just 4 times whatever term number we’re dealing with. For our fifth term which should be 20. What we’re really doing is the 5th term, term 5 times 4. Basically the term number times 4 to get the general term.
Continuing on to another example. I’m going to try the same exact approach. And in this is I want to somehow find a pattern relating to the term number. The first thing I realize is that my numerator is always 4. The numerator isn’t changing at all, so I know right away that this is going to be 4 over something. That’s easy enough. Where the harder part lies is trying to find the pattern for our denominators, and again we want to look for what these have in common.
I look at these and I see 3, 9, 27, 81, these are all powers of 3 so let’s rewrite it as such. 3 to the first, 3², 3³ and 3 to the 4th and basically what we’re doing is taking 3 to whatever number the term is. So 3 to the first for the first term, 3 to the second for the second term, so on and so forth. Our next term will hopefully be 3 to the 5th, 243 which it would be.
We know that in the denominator then we’re dealing with 3 to the n, 3 to whatever term number we’re dealing with. Again by looking for patterns, looking through breaking it down, we’re able to find the general term.
One more example. This one is a little bet less mathematical and a little bit more logical. For this what we look for is a pattern once again, and this time it’s not quite as multiplication, adding sort of family, it’s a little more logic. What we see is our first term is ½, our second term is 2/3 third term ¾, 4th term 4/5. The numerator is always the same as the term number. First term 1, second term 2, third term 3, fourth term 4, so the numerator is always the same thing as our term number, so our numerator is just going to be n.
The denominator is also related and what I see here is the denominator is always just 1 more than the numerator. You can see the relation here so if our numerator is n, what that makes our denominator is n plus 1. So our numerator is always the same as the term number, sorry, numerator is always the same as the term number the denominator is always one more than that.
The cool thing about this is you can always check your answers. Say I want to find the 4th term, a sub 4, all I would do is plug 4 into my equation, I get 4/5, first term, second term, third term, fourth term, it checks out. So there really is no excuse for getting a wrong answer because you can always plug in numbers to check your work.
Looking for patterns is the main trick for finding your general term. Whether it be figuring multiples, figuring out powers or just sort of seeing the general pattern, the main thing you want to do when finding these is just look for some sort of pattern which you can turn into an equation.