In Exercises 1–8, write the first five terms of each geometric se...

Question

In Exercises 1–8, write the first five terms of each geometric sequence.

an = - 5a_(n-1), a1 = - 6

Accepted Answer

Hello today we're going to be using the given recursive formula to find the first eight terms of the geometric sequence. So the recursive formula given to us is a sub N is equal to negative three times a sub and minus one. And we are told that the first term in the geometric sequence a sub one is negative four. So let's just talk about the recursive formula for a moment. What does it mean to find a sub and minus one while a sub and minus one refers to the previous term in any geometric sequence. So for example, when we're trying to find the second term and is equal to two, that is going to give us a sub two is equal to negative three times a sub two minus one and a sub two minus one is going to give us a sub one which is going to be the previous term in the geometric sequence. So we're going to be using this pattern to find the first eight terms of the geometric sequence. So let's go ahead and start with the second term when N is equal to two because we already know what the first term is. So when N is equal to two, we get a sub two is equal to negative three times a sub two minus one, which is just going to be a sub one and we know what a sub one is. It's negative four. So replacing one with negative four is going to give us negative three times negative four. And that's going to give us positive 12, which is going to be the second term in the geometric series. Now we want to go ahead and continue this process all the way until the eighth term of the geometric series. So next we're going to let N equal to three and when N is equal to three we get a sub three is equal to negative three times a sub three minus one which is going to be a sub two. Now we know what a sub two is, it's 12. So we're gonna replace a sub two with 12 And that's going to give us -3 times 12 which is going to give us -36. And that's going to be the third term of the geometric series sequence. Now we're gonna let N is equal to four. And when N is equal to four we get a sub four is equal to negative three times a sub four minus one which is a sub three and a sub three is negative 36. So we get negative three times negative 36 which is going to give us positive 108. Next we're gonna let N is equal to five. And when N is equal to five we get a sub five is equal to negative three times a sub five minus one which is a sub four. Sub four was 108. So we have negative three times 108 And that's going to give us -324. And that's gonna be the fifth term. Now. For the sixth term we let N is equal to six. And when n is equal to six we get a sub six is equal to negative three times a sub six minus one which is a sub five and a sub five is negative 324. So we have negative three times negative 324. And that's going to give us positive 972 For the 7th term and is equal to seven. And that's going to give us a sub seven is equal to negative three times a sub seven minus one which is a sub six. A sub six is 972. So we have negative three times 972. And that's going to give us negative 2,916. And finally for the eighth term we're going to let N is equal to eight. And that's going to give us a sub eight is equal to negative three Times a sub and a sub 8 -1 which is a sub seven. And that's going to equal to -3 times negative 2916. And that is going to equal to positive 8,748. And that's going to be the eighth term of the geometric sequence. Now let's go ahead and list out all the terms together we were told the first term was negative four. Then we calculated the rest of the seven terms, which is going to give us 12 negative 108 -324, 972 -2000, 916 and finally 8748. And this is going to be the first eight terms of the geometric sequence. And with that being said, the answer to this problem is going to be C. So I hope this video helps you in understanding how to approach this geometric sequence problem. And I'll go ahead and see you all in the next video.

In Exercises 1–8, write the first five terms of each geometric sequence. an = - 5a_(n-1), a1 = - 6

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