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# Intro to arithmetic sequences

Sal introduces arithmetic sequences and their main features, the initial term and the common difference. He gives various examples of such sequences, defined explicitly and recursively. Created by Sal Khan.

## Want to join the conversation?

• Where does n-1 come in? I didn't understand that.
• In the context of a recursive formula where we have "n-1" in subindex of "a", you can think of "a" as the previous term in the sequence. In the context of an explicit formula like "-5+2(n-1)" "n-1" represents how many times we need to add 2 to the first term to get the n-th term.
• Is there an explicit way to express the last sequence?
• Yes, there is actually an explicit formula:
a_n_=(1 + n)*n /2

Here is the proof:

The easiest way is probably to look at the sequence for a while, and realize:
a_1_=1
a_2_=1+2
a_3_=1+2+3
a_4_=1+2+3+4
---
a_n_=1+2+3+4+---+n

So we will need to add the numbers all the way from 1 to n

Intrestingly, such a formula have already been derived. I will not explain it here, but the formula is:

a_n_=n(1 + n) /2

Should you want to know how this formula was derived, take a look at http://math.stackexchange.com/questions/2260/proof-for-formula-for-sum-of-sequence-123-ldotsn

Cheers
• why is this called Arithmetic sequences?
• Good question. It's an oddly imprecise name to use for something that has a very precise definition! As far as I gather from reading up on it, it seems like there's no general consensus as to why it's been given this name. The most plausible explanation I've read (from here: https://www.math.toronto.edu/mathnet/questionCorner/arithgeom.html) is that each number in the sequence is equal to the arithmetic mean of the numbers before and after it. Overall though, it seems like a pretty bad name to choose!
• Can't we write the explicit/recursive formula this way:
Explicit: A(n)=k+d(n-1)
Recursive: A(1)=k, A(n)=A(n-1)+d
d is the common difference and k is the initial value
Can't we write it that way where A is a function of the step 'n'?
• Why is he pronouncing arithmetic air-eth-matic?
• Uh-rith-muh-tic is a noun. Air-eth-matic is an adjective.
• why do you put an-1? when you don't even subtract 1 at all when using the recursive formula.
• This still confuses me and I don't understand any of it. does anyone have a different way of explaining it?
• An arithmetic sequence is a sequence where each term increases by adding/subtracting some constant k.
This is in contrast to a geometric sequence where each term increases by dividing/multiplying some constant k.

Example:
a1 = 25
a(n) = a(n-1) + 5

Hope this helps,
- Convenient Colleague
• Can you add negative numbers, like -6, with arithmetic sequences?
• Yes. An arithmetic sequence is any sequence with a constant difference between the terms. The terms themselves (and/or the sum) can be positive, negative, or zero.