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

an = - 4a_(n-1), a1 = 10

Question

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

an = - 4a_(n-1), a1 = 10

Accepted Answer

Hello today, we're going to be using the given recursive formula to find the first eight terms of a geometric sequence. So the recursive formula given to us is a sub N is equal to two times a sub and minus one. And we are told that the first term is five. Now for the recursive formula. Whenever we want to look for a term saying the second term, for example, when N is equal to two, we're going to replace N with two. And so that's going to give us a sub two is equal to two times a sub n minus one, which refers to the previous term in a geometric sequence. That's going to give us two times a sub two minus one and a sub two minus one is going to give us a sub one. So finding a sub two is going to give us two times the first term, a sub one. And this is going to be the pattern that we're going to need to use in order to find the next seven terms in the geometric sequence. So, let's go ahead and start by finding the second term of the sequence when n is equal to two, When n is equal to two, we get a sub two is equal to two times a sub n minus one, which is just a sub one. An ace of one refers to the first term in the geometric sequence which we know what it is. It's five because it's been given to us. So we're gonna replace ace of one with five. And that's going to give us two times five which is equal to 10 and this is going to be the second term in the geometric sequence. Next let's go ahead and find the third term which is one. N is equal to three. When N is equal to three we get a sub three is equal to two times a sub three minus one which is a sub two. Now we know what a sub two is. We calculated it in the previous step, it's 10. So we're gonna replace a sub two with 10 and that's going to give us two times 10 which is equal to 20. And we're going to repeat this process all the way until we get the eighth term. So next we're gonna let N is equal to four And we're going to get a sub four Is equal to two times a sub 4 -1 which is a sub three and a sub three is equal to 20. So we have two times 20 which is equal to 40. Next we're gonna find the fifth term which is N is equal to five. And when n is equal to five we have a sub five is equal to two times a sub five minus one which is a sub four And a sub four is 40. So we have two times 40 which is going to give us 80. Next we're gonna let N is equal to six. And when n is equal to six we have a sub six is equal to two times a sub six minus one, which is a sub five, A sub five is 80. So we have two times 80 which is going to give us 160. Next we let N is equal to seven and we get a sub seven Is equal to two times a sub 7 -1, which is a sub six And a sub six is 160. So we have two times 160 which is going to give us 320. And finally we're going to let N is equal to eight. And when N is equal to eight we get a sub eight is equal to two times a sub eight minus one, which is a sub seven and that is going to equal to two times 320 And that's going to equal to 640. So we have the first eight terms of the geometric sequence. Let's go ahead and list them out. So we were told the first term is five, then we calculated the remaining seven, which turned out to be 10, 40, 80, 160, 320 And 640. 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 a 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 = - 4a_(n-1), a1 = 10

Verified Solution

Key Concepts

Geometric Sequence

Recursive Formula

Initial Term