Geometric Sequences - Concept
When modeling real world situations, we often use what's called inverse or indirect variation to describe a relation between two variables. Indirect variation is a relation in which the absolute value of one variable gets smaller while the other gets larger. Indirect variation and direct variation are important concepts to understand when learning equations and interpreting graphs.
A geometric sequence is a series of numbers where basically going from one to the next [IB] we are multiplying by a constant rate, okay? So right behind me what I have is, we're going from 2 to 6, multiplying by 3 from 6 to 18 we're multiplying by 3 as well. Okay? So that 3 is consistent so therefore I know I have a geometric sequence. Okay.
If this was just 2 terms, I wouldn't actually know what's going on because I could go from 2 to 6 one of 2 ways. I could either add 4 which would be an arithmetic sequence or multiply by 3 which tells me it's geometric. So I need that third term in order to figure out what exactly ki- what kind of sequence it really is, okay?
What we're going to do now is find the general term, find the s sub n in order for a geometric sequence, okay? So we're given the first term. s of 1 is equal to the first term. And in order to find the second term, what we do is take that first term and multiply it by that common rate okay and then we call it r.
To find the third term then we take the second term and multiply it by the rate as well okay. So we went from 2 to 6 and then [IB] 6 to 18 we again multiplied it by that common rate 3. But we know that a sub 2 is actually a sub 1 times r. So what we end up with is this is equal to a sub 1 times r squared. Continuing down the row, a sub 4 is just going to be another rate times the previous term leaving us with a sub 1 times r cubed, so on and so forth. Eventually we're going get to our general term. Ace of n is equal to ace of 1. And if you notice there's always one less rate than the term number, okay. So our second term has one rate, third term has 2 rates, fourth term has 3 rates. So if we're dealing with ace of n we are just going to have r to the n minus 1.
So, that is an introduction to a geometric sequence and the general term for geometric sequence ace of n is equal to ace of 1 times r to the n minus 1.