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### Course: Operations and Algebraic Thinking 229+>Unit 7

Lesson 1: Introduction to arithmetic sequences

# 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.
• Yes, the Fibonacci numbers are an example of a sequence.

Have a blessed, wonderful day!
• WHAT ARE WE REFERING TO AS K here
• k is the index/position of a term in the sequence. for example, the 1st term in a sequence would have a k value of 1, the 2nd term would have a k value of 2, the 34th term would have a k value of 34, and so on
another example: in the sequence, {1, 4, 7, 10}, 4, being the 2nd term in the sequence, has a k value of 2.
• 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.
• 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.
• a sub k seems to be the way to read aloud ak as it is written at , ie with the k below the a (ie k is the 'subscript'). I will use this 'a sub k', as I can't work out how to make subscripts.
Explicit seems to mean either writing out the numbers directly, or providing a rule(function) for how to get any term in the sequence. For example if 'a sub k' is 2k, you can directly calculate 'a sub 5' would be 10. Something is recursive is when in order to use the rule to work out the term, you need the value of the term before. So if you used the rule
'a sub k' = 'a sub (k-1)' + 2
you couldn't directly calculate 'a sub 5'. You would need to know 'a sub 4'. But to calculate that, you need 'a sub 3', but for that .... etc. This is 'recursive'
• `k-1` is the number of differences that have to the be added to the first term to get the kth term. For example, if you want to get the 2nd term, you add 1 difference to the first term, and if you want the 3rd term, you add 2 differences to the first term (or 1 difference to the 2nd term), and so on.