Write a formula for the general term (the nth term) of each geome...

Question

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a(sub n) to find a(sub 8), the eighth term of the sequence. 1, 2, 4, 8, ...

Accepted Answer

Welcome back. I'm so glad you're here. We've got a sequence. 139 27 etcetera. And we want to find a formula for the seventh term of the sequence. And the value of that seventh term or the value of a sub seven. So what kind of sequence do we have? We go from 1-3 and then three jumps all the way up to nine. Well that's not arithmetic. There isn't a constant term being added to those numbers to go from one to the next. So is it geometric? We can figure that out by looking to see the ratio. We're going to do a couple of test points to see if the ratio is the same between some subsequent terms. So we can pick two subsequent terms. I'll take the first and the second and we take the second one that we choose. So I'll say a sub two divided by the first one, right proceeding in front of it. So a sub two divided by a sub one. Or in other words three divided by one and that is three. Now we can test it to see if the ratio is the same with two other subsequent points. This time we'll take a sub three and the term immediately preceding it A sub two. In other words, ace of three. The third terms value is nine. A sub two was three. So nine divided by three. That is also three. So we do have a geometric sequence and the ratio for this sequence is three. When we know we have a geometric sequence we have a formula for the nth term of that sequence. We recall from previous lessons that that formula is a sub N equals a sub one. Times are Raised to the N -1. And what is N exactly N. Is that 10th term? The number of the term that we're looking for this time? It's seven, so N equals seven in this case and we did sell for our already are is that ratio? And our is three. So we can fill those numbers in A sub seven equals a sub one times To the 7 -1. And we can simplify that exponents. So we've got a sub seven equals a sub one times three rays to the sixth. That's the formula that we were looking for. Now let's solve for the value of the seventh term. We do know what a sub one is. A sub one is the value of the first term. So ace of one equals one and filling that in. We've got a sub seven equals one times three. Race to the six. So now we're just evaluating for three raised to the sixth and that is 729. We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one

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