>> Here is another one. Now this one involves fractions, and we may be told here--Let's presume that we're told that it's a geometric sequence, and we wanna find a particular item, A sub 8. Now what's not clear here perhaps is the value for R, the common ratio. What are we multiplying by to get the succeeding terms here? Now we can kind of use trial and error to see through it, or we can use another kind of technique. Now we know this much though, we know that A sub 1 is negative 1/3. Okay, good. Now, what about the ratio? The common ratio is the ratio of succeeding terms. That is I could take something like A sub 3 and divide by A sub 2 to find that common ratio. Let's do that here. This is, let's see, A sub 3 is negative 1 over 27 and A sub 2 is 1/9. So, I could make this calculation, you see. Now it's a complex fraction and we know that it's negative 1 over 27 divided by 1/9 which means negative 1 over 27 times 9 over 1. Oh, happy day, we have a cancellation, the 9 and the 9, 1, 9 and the 27, 3. So we get negative 1/3 for the common ratio. So that's the way to make the calculation if we're ever confused. Okay. Negative 1/3 then is the common ratio. So, we can write the expression for the nth term. The nth is first term times R to the N minus 1, and away we go to find the eighth term, A sub 8. And we have all of the information we need. Let's see, the first term from right here is 1/3. The common ratio is negative 1/3. Excuse me, the first term is negative 1/3, my bad. Common ratio is negative 1/3, and we want the N minus 1. Well, that's 8 minus 1, you see, we want the eighth term. So it would be 8 minus 1 which is 7. So we have A sub 8 then is negative 1/3 times negative 1/3 to the 7th. But if these two are alike, then I have like bases, I can add the exponents, the exponent here is understood to be 1, so we could write this as negative 1/3 to the eighth power. And there are other ways to manipulate this as well. I mean we could pop it into the calculator right now but just realize that there are a lot of ways to manipulate fractions raised to exponents like this. I'm noticing first of all that I've got an even exponent on a negative number. That's gonna make this value positive. So I can eliminate the negative sign right away. I could also apply this exponent to numerator and denominator separately and I could think of this as 1 over 3 to the eighth power if I wanted to. Now it's important to realize these different manipulations because you may have an answer in the back of your textbook that is some variation on this manipulation and you have to be able to recognize whether or not you have correct answers or not. And by the way, 3 to the eighth power is 6,561. So you might see that in this denominator. And if we wanted a decimal approximation, we could pop it into the calculator.