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

Lesson 1: Introduction to arithmetic sequences- Sequences intro
- Intro to arithmetic sequences
- Intro to arithmetic sequences
- Extending arithmetic sequences
- Extend arithmetic sequences
- Using arithmetic sequences formulas
- Intro to arithmetic sequence formulas
- Worked example: using recursive formula for arithmetic sequence
- Use arithmetic sequence formulas

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# Intro to arithmetic sequences

Sal introduces

*and their main features, the***arithmetic sequences***and the***initial term****. He gives various examples of such sequences, defined explicitly and recursively. Created by Sal Khan.***common difference*## Want to join the conversation?

- Where does n-1 come in? I didn't understand that.(152 votes)
- 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.(49 votes)

- Is there an explicit way to express the last sequence?(27 votes)
- 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(55 votes)

- why is this called Arithmetic sequences?(13 votes)
- 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!(24 votes)

- Why is he pronouncing arithmetic air-eth-matic?(4 votes)
- Uh-rith-muh-tic is a noun. Air-eth-matic is an adjective.(18 votes)

- 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'?(8 votes) - why do you put an-1? when you don't even subtract 1 at all when using the recursive formula.(2 votes)
- From what I understand, 'n' stands for your index, or counter, variable. In the recursive formula Sal showed us, 'a_n', the number you are solving for, is equal to the value of the previous number, or 'a_n-1' PLUS the value of the index itself. You take the value of the previous number and add 'n' to it. The -1 part of the 'n-1' subtracts 1 from your index, 'n', to give you the number to which you add 'n'. I hope this helps!(1 vote)

- This still confuses me and I don't understand any of it. does anyone have a different way of explaining it?(4 votes)
- 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(6 votes)

- Can you add negative numbers, like -6, with arithmetic sequences?(4 votes)
- 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.(4 votes)

- can an arithmetic sequence start with 0? For example; {0,2,4,6,8...}(2 votes)
- Of course it can ^^(5 votes)

- So is it safe to think that a sequence is like a function with a domain?(2 votes)
- Yes, it's more than safe. A sequence is defined as a function whose domain is the natural numbers.(5 votes)

## Video transcript

What I want to do in this
video is familiarize ourselves with a very common
class of sequences. And this is
arithmetic sequences. And they are usually
pretty easy to spot. They are sequences where each
term is a fixed number larger than the term before it. So my goal here is to figure
out which of these sequences are arithmetic sequences. And then just so that
we have some practice with some of the
sequence notation, I want to define them
either as explicit functions of the term you're looking for,
the index you're looking at, or as recursive definitions. So first, given that
an arithmetic sequence is one where each successive
term is a fixed amount larger than the previous one, which of
these are arithmetic sequences? Well let's look at this
first one right over here. To go from negative 5 to
negative 3, we had to add 2. Then to go from negative 3 to
negative 1, you have to add 2. Then to go from negative
1 to 1, you had to add 2. So this is clearly an
arithmetic sequence. We're adding the same
amount every time. And there are several ways that
we could define the sequence. We could say it's a sub n. And you don't always
have to use k. This time I'll use n
to denote our index. From n equals 1 to infinity
with-- and there's two ways we could define it. We could either
define it explicitly, or we could define
it recursively. So if we wanted to
define it explicitly, we could write a sub n is equal
to whatever the first term is. In this case, our first
term is negative 5. It's equal to negative
5 plus-- we're going to add 2 one less
times than the term we're at. So for the second
term, we add 2 once. For the third term,
we add 2 twice. For the fourth term,
from our base term, we added 2 three times. So we're going to add 2. We're going to add positive
2 one less than the index that we're looking
at-- n minus 1 times. So this is an
explicit definition of this arithmetic sequence. If I wanted to write
it recursively, I could say a sub 1 is
equal to negative 5. And then each successive term,
for a sub 2 and greater-- so I could say a sub n is equal
to a sub n minus 1 plus 3. Each term is equal to the
previous term-- oh, not 3-- plus 2. So this is for n is
greater than or equal to 2. So either of these
are completely legitimate ways of defining
the arithmetic sequence that we have here. We can either define
it explicitly, or we could define
it recursively. Now let's look at this sequence. Is this one arithmetic? Well, we're going from 100. We add 7. 107 to 114, we're adding 7. 114 to 121, we are adding 7. So this is indeed an
arithmetic sequence. So just to be
clear, this is one, and this is one right over here. And we could write that this
is the sequence a sub n, n going from 1 to infinity
of-- and we could just say a sub n, if we want
to define it explicitly, is equal to 100 plus
we're adding 7 every time. And then each term-- the
second term we added 7 once. Third term-- we add 7 twice. So for the nth term, we're
going to add 7 n minus 1 times. So this is an explicit
definition of it, but we could also
do it recursively. So just to be clear, this is
one definition where we write it like this, or we
could write a sub n, from n equals 1 to infinity. And in either case
I should write with. And if I want to
define it recursively, I could say a sub
1 is equal to 100. And then, for anything larger
than 1, for any index above 1, a sub n is equal to the
previous term plus 7. And so we're done. This is another
way of defining it. So in general, if you
wanted a generalizable way to spot or define an
arithmetic sequence, you could say an
arithmetic sequence is going to be of the form
a sub n-- if we're talking about an infinite one--
from n equals 1 to infinity. If you want to
define it explicitly, you could say a sub n is
equal to some constant, which would essentially
the first term. It would be some
constant plus some number that your incrementing--
or I guess this could be a negative
number, or decrementing by-- times n minus 1. So this is one way to define
an arithmetic sequence. In this case, d was 2. In this case, d is 7. That's how much you're
adding by each time. And in this case, k is negative
5, and in this case, k is 100. The other way, if you
wanted to the right the recursive way of defining an
arithmetic sequence generally, you could say a sub
1 is equal to k, and then a sub n is
equal to a sub n minus 1. A given term is equal
to the previous term plus d for n greater
than or equal to 2. So once again, this is explicit. This is the recursive
way of defining it. And we would just
write with there. Now the last question I have
is is this one right over here an arithmetic sequence? Well, let's check it out. We start at 1. Then we add 2. Then we add 3. So this is an immediate
giveaway that this is not an arithmetic sequence. Now we are adding 4. We're adding a different
amount every time. So this, first of all,
this is not arithmetic. This is not an
arithmetic sequence. But how could we define
this, since we're trying to define our sequences? Let's say we wanted to
define it recursively. So we could say, this is
equal to a sub n, where n is starting at 1 and
it's going to infinity, with-- we'll say our base
case-- a sub 1 is equal to 1. And then for n is 2
or greater, a sub n is going to be equal to what? So a sub 2 is the previous
term plus 2. a sub 3 is the previous
term plus 3. a sub 4 is the previous term plus 4. So it's going to be the
previous term plus whatever your index is. So this looks close,
but notice here we're changing the amount
that we're adding based on what our index is. We're adding the amount of
index to the previous term. And so this is for n is
greater than or equal to 2. Well for an arithmetic
sequence, we're adding the same
amount regardless of what our index is. Here we're adding
the index itself. So this one is not
arithmetic, but it's an interesting
sequence nonetheless.