Worked example: sequence explicit formula | Series | AP Calculus BC | Khan Academy

If a_sub_n is equal to (n^2 - 10) / (n + 1), determine a_sub_4 + a_sub_9.

Well, let's just think about each of these independently.

a_sub_4, let me write it this way: a the fourth term. So a_sub_4, so our n, our lowercase n, is going to be four. It's 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 (4^2 - 10) / (4 + 1).

Which is equal to, well, let's see, that's (16 - 10) / 5, which is equal to 6 / 5.

So that is a_sub_4. That is the fourth term.

Now let's think about a_sub_9.

So a_sub_n, 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 9, or we're looking at the ninth term.

So it's going to be (9^2 - 10) / (9 + 1).

Let that blue color just so we see what we're doing: (9^2 - 10) / (9 + 1).

Is equal to, on the numerator we have (81 - 10), over (10).

(9 + 1).

And so this is going to be equal to 71 / 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_4 is (6/5) plus a_sub_9 which is (71/10).

Well, we can rewrite (6/5) as being equal to (12/10).

(12/10) and then (71/10) so plus (71/10), 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.