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Worked example: sequence explicit formula

Using the explicit formula of a sequence to find a couple of its terms. Created by Sal Khan.

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Video transcript

- [Instructor] If a sub n is equal to n squared minus 10 over n plus one, determine a sub four plus a sub nine. Well, let's just think about each of these independently. A sub four, a, let me write it this way. A, the fourth term, so a sub four, so our n, our lowercase n, is going to be four, is going to be equal to everywhere we see an n in this explicit definition for this sequence. Everywhere we see an n, we would replace it with a four. So it's going to be equal to four squared minus 10 over four, over four plus one, over four plus one, which is equal to, well, let's see, that's 16 minus 10 over five, which is equal to six over five. So that is a sub four. That is the fourth term. Now let's think about a sub nine. So a sub nine, so, once again, everywhere that we see an n, we would replace it with a nine. We're looking at when lowercase n is equal to nine, or we're looking at the ninth term. So it's going to be nine squared. Let me do that blue color just so we see what we're doing. Nine squared, we'll do it in a green color, minus 10 over nine plus one, over nine plus one is equal to, well, in the numerator, we have 81 minus 10, 81 minus 10 over 10, over nine plus one. And so this is going to be equal to 71 over 10. Now they want us to sum these two things, so that's going to be equal to, it's going to be equal to 6/5. A sub four is 6/5, plus a sub nine, which is 71 over 10. 71 over 10. Well, we can rewrite 6/5 as being equal to 12/10, 12/10 and then 71/10, so plus 71 over 10, which is equal to, well, if I have 12/10 and then I have another 71/10, now I'm going to have 83/10, 83/10. And we're done.