In Exercises 51–56, the general term of a sequence is given. Dete...

Question

In Exercises 51–56, the general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio.

an = n^2 + 5

Accepted Answer

Hello everyone in this video we're going to be looking at this practice problem where we're going to be looking at the sequence A Is equal to end to the 3rd plus one. So in this case we want to try to see if the sequence is an arithmetic sequence and if it is determined the common difference, if it's a geometric sequence determine the common ratio. So in order to see what sort of sequence it is, let's go ahead and write out some of its terms. So when N is equal to one, that is the first term. So we have 12 the third plus one. So we have one plus one or two. So the first term is to if I have the second term or when N is equal to two we have two to the third plus one and two to the third is two times 24 times two again. So eight plus one or nine. And if we have the third term we have three to the third plus one. So three to the third is three times three or nine and another three. So nine times 3 27 plus one. And that is equal to 28. So let's do one last term when n is equal to four or the fourth term, we have four to the third plus one and that is equal to four times four, which is 16 times four, That equals 64 plus one. So 65. So looking at these numbers 29, 28 65 I'm not seeing a common difference or a common ratio but we can go ahead and try. So if we were to look at a common difference I'll do the second term minus the first term. So 9 -2. That means that the common difference between those two terms is seven. So I should be able to add seven to my second term and get my third term. But if I add 7-9 I get 16 and not 28. So I know that it's not an arithmetic sequence. Now let's see if we can find a common ratio. So I'll denote the common ratio as our and similar to when I was looking for a common difference. I'll take my second term nine and divided by my first term too. And that gives me 4.5. So if I multiply my second term by 4.5 I should get my third term. However if I multiply nine times 4.5 I got 40.5 and not 28. So this tells me that I don't have a geometric sequence either which means that A. N. Is equal to enter the third plus one is neither arithmetic or geometric. So hopefully this video helped and see you next time

In Exercises 51–56, the general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. an = n^2 + 5

Watch next