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### Course: AP®︎ Calculus BC (2017 edition) > Unit 12

Lesson 1: Infinite sequences# Worked example: sequence explicit formula

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

## Want to join the conversation?

- What is differnce between sequence and series?(13 votes)
- A series is the sum of the members of a sequences.

Thus, the sequence is just the set of numbers. The series is what you get when you add all of those numbers together.(28 votes)

- when would a real world problem be modeled by this kind of equation?(5 votes)
- We'd probably have to stretch a bit to come up with a plausible real world problem quite like this. However, that's true for many problems we work through in calculus. In the real world you'll never be faced with the problem of cutting a wire into two pieces such that you can maximize the total area of a triangle made from one piece and a square made with the other, for example, but you may deal with that question in learning about max/min problems. The point of the problem is to develop skill in handling a concept or procedure that will be useful as we move on to other concepts and procedures (in this case, building an understanding of sequences that will help us study series, which are incredibly useful).(10 votes)

- Why does the sequence not start at n = 0 so that for a sub 4, the n= 3 instead of 4?(4 votes)
- Usually, you start at n=1 to get the first term.

You could start generating terms from n=anything, restricted only by the domain of your function. Rarely, though, do you care about the zeroth term, or the negative eighth term (even though you could find them if you really wanted to).

As you progress with sequences and series, you will come to places where it is important that you follow the instructions for which terms to use.(2 votes)

- what is finding terms explicitly defined sequence good for(2 votes)
- First thing is it is a way of making sure you really understand some of the properties of sequences. Later in this course you will need to figure out the nth term of the derivative of a Taylor polynomial.

Finally, also with respect to Taylor polynomials, when we use them to estimate the value of a function, the first term not used is our error bound, so we need to know the value of that term.

There may be more good reasons, but these are the first three off the top of my head.(3 votes)

- how come 6/5 change to 12/10 in what way does it change can you explain ?(2 votes)
- At .29 why are you replacing n with four instead of any other numbers?(1 vote)
- Because the question asks for a4 + a9. So he is finding a4, which means "replace n with 4 in the formula for an".(2 votes)

- Can we have negative "n"'s? Like a sub -2? Instead of a positive integer...

But if we do wont we come to an undefined answer? n=-1 wont we have a 0 in the denominator?(1 vote) - i would to know how to find a geometric progression if any 2 terms are given?(1 vote)
- What is the formula of finding sequence?(1 vote)
- How do we solve this 2An-1 +1, A0=3(1 vote)
- If it's an arithmetic sequence, then we have 𝐴(𝑛) = 𝐴(0) + 𝑛𝑑 = 3 + 𝑛𝑑, where 𝑑 is the difference between consecutive terms.

So, 2∙𝐴(𝑛 − 1) + 1 = 2(3 + (𝑛 − 1)𝑑) + 1 = 7 + 2(𝑛 − 1)𝑑(1 vote)

## 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.