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## Calculus, all content (2017 edition)

### Course: Calculus, all content (2017 edition)>Unit 7

Lesson 1: Sequences review

# Sequences intro

Sequences are ordered lists of numbers (called "terms"), like 2,5,8. Some sequences follow a specific pattern that can be used to extend them indefinitely. For example, 2,5,8 follows the pattern "add 3," and now we can continue the sequence. Sequences can have formulas that tell us how to find any term in the sequence. For example, 2,5,8,... can be represented by the formula 2+3(n-1). Created by Sal Khan.

## Want to join the conversation?

• What is the difference between finite and infinite sequence, as they both have similar functions? • In a finite sequence, there are a limited number of values for k. For example, the first finite sequence that Sal lists has values of k from 1 to 4. Because there are four values of k, the sequence only contains 4 numbers and is therefore finite.
Infinite sequences, on the other hand, contain an unlimited number of values for k. The first example of an infinite sequence that Sal lists has values of k from 1 to ∞. Because there are infinite values of k, the sequence contains infinite numbers and is therefore infinite.
• So, what is the difference between a function and a sequence? Why is a sequence discrete and a function is continuous? •  In general, a function is a relations that defines an output for any input over an interval. Thus, you are often able to visualize this set of outputs on a graph as a continuous line.

A sequence, on the other hand, is a relation that defines an output only for integer inputs. Because you cannot get an output for any value in between, you can only visualize the set of outputs on a graph as a set of discrete points.
• I don't get what recursive is, can someone explain it please? • In a recursively defined sequence, the next term is defined in terms of (excuse the pun) the terms that come before it. For example you could have a sequence where the first term is equal to 1 and where each term that follows is equal to the sum of all the previous terms: 1, 1, 2, 4, 8, 16,...
• Why is it always -1 in the end of the equation in the parenthesis? Is it possible for it to be a different number? • We have the (k-1) multiplied to the common difference so that the formula is valid for all terms, including the first term. The first term(k=1) does not have the common ratio added to it. So for the first term (k-1) will become 0 since k=1. This is why there is (k-1) in the general formula.
However, if the first term is divisible by the common difference, the k-1 can be changed to some other factor using algebraic manipulation. The formula remains the same but we only change or simplify the way we write it.
For example take the sequence 2,4,6,8.....
Its general formula is-->
`t(k)=2+2(k-1)`
However if we open the bracket we get-->
`t(k)=2+2k-2=2k`
So the (k-1) factor will always be there in the general formula but in some cases we can simplify the formula to get a different form.
• Are the Fibonacci numbers considered a sequence?
1,1,2,3,5,8,13,21 and so on. • What is the difference between DENOTING a sequence and DEFINING a sequence? Are explicit and recursive formulas denotations or definitions? • "Define" a sequence is the act of establish a law who's govern a sequence. Like the arithmetic sequences in the video (one with the law +3 in each previous term of the sequence, and another with +4 in each previous term of the sequence). "Denoting" means showing something. Usually with a especific set of simbols and notations. Like Sal shows in the video, how do you express a sequence, using a regular notation or a function notation.
• at , why does he use k-1, why is that needed? • What is the clear distinction between a sequence and a function?
Is it the domain or any other characteristic that distinguishes? • I would say that a sequence is a special kind of function that has the natural numbers, ℕ, as its domain. I think you'd also have to say the domain was either infinite (defined for all natural numbers) or, if it's finite, then it's defined for the first n natural numbers. In other words you can't have gaps.
A sequence definitely satisfies the requirement that functions be one-to-one.
• For those struggling with:

`k-1`

This is essentially a "hack" to avoid counting your current "index" location against the math. For example:

`A(k) = 1+3(k)`

PreNote: ( k=1 is an index location, like finding a book in a library. This number increments each time across the "loop" and can be seen as similar to the Sigma∑ notation's looping functionality in that respect.)

If We look at K=1 and did not subtract 1 from the current index we would actually get 1+3(1) = 4 or 1+3(2)=7. That doesn't make sense to what we want. The current Index can be seen as offset by 1 due to starting at 1. We need to subtract 1 to bring back that balance. 1+3(1-1) = 4, 1+3(2-1) = 4, 1+3(3-1) = 7.

In modern Computer Science(Programming), we don't work with Indexes like this any longer, and starting an Index at 1 is generally fallen out of fashion largely in part of this constant need to work around the problem. You can easily avoid this problem in your own work by explicitly starting your K to start at 0.

I assume for quizzes however that they will continue to specify the start 1, so just work around it. • I don't understand some of the words being used.
What does recursive, "a sub k" and explicit mean?
Please use less complex words when explaining math problems. 