#help-28
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From the graph, you can see the local minimum and local maximum
I havent done this before can you elaborate
I need to find out the minimums too
@mighty crescent
Do you know what local minimum and local maximum mean?
no
Local maximum is the left one and local minimum is the right one
You substitute into the expression for the function
the maximum is over (-1,0) and then I plug in -1 to the function to get the second number
ty ty
what does this mean? @mighty crescent
the maximas are y outputs. You'd find the value by plugging them into the equation above.
In this case something like this
,w 2(-1)^3 - 6(-1)^2 - 18(-1) + 7
so that's your relative maximum
not really, that's just the graphing window. It's says that the x values are from -5 to 5 with each tick being 1 unit and the y values are form -60 to 60 with each tick being 10 units
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where am i wrong ?
@plain fox Has your question been resolved?
<@&286206848099549185> pls help
Shine some light or retrace your work, it's difficult to comprehend.
it is clear

i think u can use agp here
whats agp?
arithmetic-geometric progression
no i need to use binomial
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u mean using quadratic formula?
or by completing the square
but that i cant use that , i have to use binomial
if i could use those methods , this question was a piece of cake
but i have to use the binomial and stuff
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can anyone solve?
close either this or your previous channel.
they are both different questions
i'll ask this later after the prev is solved then
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@atomic kayak Has your question been resolved?
@atomic kayak Has your question been resolved?
<@&286206848099549185>
yess
please read #❓how-to-get-help
@atomic kayak Has your question been resolved?
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Hey
What am I missing here? how did she get to this?
It's by definition
what happened to m+1?
n+1 can be written as (n + m + 2)-(m+1)
Got rewritten as (n + m + 2) - (n + 1)
If you still don't understand, try rewriting the right hand side using the definition (a choose b is equal to a!/((a - b)!b!))
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how do i solve this
can you help me
Express P_n(-x)
Right, which is the same as (-1)^k x^k
The two (-1)^k's will combine into just 1
So you have $P_n(-x) = \sum_{k=0}^n\frac{1}{k!}x^k$
A Lonely Bean
what happened to the k!
?
oh
im kind of confused
By what exactly?
are u saying this
is equal to this?
Yes, because (-1)^k * (-x)^k = (-1)^k * (-1)^k * x^k = (-1)^2k * x^k = ((-1)^2)^k * x^k = 1^k * x^k = x^k
Now it looks like you need to use the formula for multiplying power series together
what is the formula
$\left(\sum_{k=0}^\infty a_kx^k\right)\left(\sum_{k=0}^\infty b_kx^k\right) = \sum_{k=0}^\infty\left(\sum_{i=0}^na_ib_{n-i}\right)x^k$
omg!
amen!
A Lonely Bean
So in this case we have a_k = (-1)^k/k! and b_k = 1/k!
Hold on P_n isn't a power series, it's a polynomial
amen...this isnt calculus
Wait is this a thing
Well the formula for multiplying polynomials is not so different, it's just
[ \left(\sum_{k=0}^n a_kx^k\right)\left(\sum_{k=0}^nb_kx^k\right) = \sum_{k=0}^{2n}\left(\sum_{i=0}^ka_ib_{k-i}\right)x^k ]
As far as I remember
💡
Yes, multiplication of power series is defined using the convlution too
What pattern
I'll show later why that works
A Lonely Bean
Anyway here we are gonna have a_k = (-1)^k/k! and b_k = 1/k!
So $P_n(x)P_n(-x) = \sum_{k=0}^{2n}\left(\sum_{i=0}^k\frac{(-1)^i}{i!}\frac{1}{(k-i)!}\right)x^k$
A Lonely Bean
Hmm
amen
It looks like we need to consider the cases of k being even or odd
If k is odd, then every ith term gets cancelled with the (k - i)th term
So we look at only even k
I.e., $P_n(x)P_n(-x) = \sum_{k=0}^n\left(\sum_{i=0}^{2k}\frac{(-1)^i}{i!(k-i)!}\right)x^{2k}$
A Lonely Bean
ok
Now you may notice that every ith term in that sum is equal to the (2k - i)th term, we can add them up
Except for the central term, since it is not paired with a term equal to it (since i = 2k - i for that term)
ok
Eh this doesn't seem to be going anywhere
I'll let you know if I come up with the solution
ill ask my professor
Oh it's you
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Hey guys. I need help on how to calculate this
Are you aware of sine law
no
if two triangles are equal then ratio of corresponding sides should be equal
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I want to sort n marbles into n/m bags of m marbles each. The marbles are colored, and there are c(i) marbles of color i. Bags cannot contain more than 1 marble of each color.
How many ways can I do this?
I can do it ignoring colors, but after a while I haven't found a viable solution to exclude color repetitons besides brute force. And I'm dealing with around 50 marbles, so that is not possible.
This is not related to any study or work, so I don't even know if it's possible to find a simpler solution.
@lucid hill Has your question been resolved?
@lucid hill Has your question been resolved?
@lucid hill Has your question been resolved?
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I have this math problem that looks like this. I have to find both solutions for x. I was able to find x = 3 but it says the other solution is x = 7 and I’m not sure how they got that answer. Can anyone explain?
@rare dock snow
snow
Square both sides, isolate the square root, then square again
Could you show your work?
Sure I’ll show what I did to get 3
So here’s how I got 3 and I checked it at the bottom
riemann
But also you didn't square the right side
Oh when I do that I get x = 4 which is wrong
Right. It's wrong for two reasons
But somehow they cancelled each other and got the right answer
Go back to this
(a+b)^2 = ?
I should end up with 5x+1 - 3x-5 = 4 right?
@lucid void Has your question been resolved?
@lucid void Has your question been resolved?
No
You're still making this mistake
Answer this
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I got this question incorrect, can someone tell me which one of these was the correct answer, and why that is the case?
The slices are perpendicular to the y axis, so I believe I would take the integral with respect to x, and since it is a square, I think the correct answer would have been the one above what I chose.
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hello, could someone explain me this please, is about variations in combinatorics, without repetition
I don't get (n-k+1)
the other part I get it
it's basically multiplying k numbers starting from n and going down
what do you mean?
mm w8
I think there should be a parenthesis like this ((n-k)+1) or is it not necessary?
not necessary, they mean the same thing
(n-k)+1 is the same as n-k+1
what is your point
because you didnt apply the formula well
why
and whats the formula saying ?
(n-k+1)
before that
n (n-1)(n-2)......(n-k+1)
?
n-(k+1) = n-k-1
not the same thing with your formula
dont put parenthesis where there is none
or at least know how it works
oh okay, I calculated it like this n-(k+1)
I don't have teachers
sadge
thanks so much 🙂 I made this mistake that now I will pay attention to, sorry for this stupid error
no problemo
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For B why doesn’t reverse chain rule work?
1b?
reverse chain rule?
Is that integration by parts?
it would be u-substitution
Yes I’m supposed to use that
integration by parts is reverse product rule
U can apply ibp many times, not only once
ye we'll need to see more work to understand what you're trying to do
Ok
Why does the method I use work on the right but not for the one on the left
@brittle steeple @kind jay
i don't know what your method is
first you have written
y = e^(6x^2-1)
and then later you have written
y = 1/2 * e^(6x^2-1)
you are aware these are different things, right
so y cannot equal them both
@slender pond Has your question been resolved?
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Hi, this a practice question and I need help in finding the domain and range with X intercept and y intercept, this might be easy but I’m stuck for the domain is it (0, infinty)?
no
why would domain be (0,inf)? 
the domain is all of the x-values you can put into the function
looking at this, can you put in x-values of 0 or below?
Yes I think so
you think so? or yes
Yes
why
Wait so this graph the thing I’m confused with is it does not touch the X axis I think bcos both the arrows are in y axis as I see it, but I know I’m wrong
So if it doesn’t touch the X axis at all and the domain should be the values of X axis, does this mean the domain is 0?
nope
Oooo so it’s all the values
domain has nothing to do with the x-axis
I see
the domain is the set of all x-values you can input into your function
remember what functions do
they take something in (inputs)
and spit something out (outputs)
True
so the domain is all of the possible inputs
an easy example is something like
y=x
what can you put into this?
is there any limitations on what x is?
so the domain is (-inf, inf)
Ooo
1/c is not a fixed value so it depends on the value of X
So I think we can
Yes
We can
It’s 1
1/0 = 1?
1/0 = 0?
Wait so no result?
I see
No
And 0 cannot be domain
Yes
x can be 0
0 is in the domain of y=x
but for y=1/x
it is not
it depends on the function
so 0 is in the domain of some functions, and not in the domain of some other functions
makes sense so it depends on the function
Yes
Got it
Rinz let me show you a quick example of something
Yes please
Austin
what is the solution of this?
2x= X, so 2x-x = -x , x=0?
Austin
Austin
True
well this is clearly not true
so how did we get this false statement?
can we not divide by x?
well
since x is 0
you can't divide by 0
otherwise
2=1
and other bad things
so how about you try this, what is the domain of $f(x)=\frac{1}{1-x}$
Austin
what x-values can you not put in?
The 1-x cannot be equal to 0
which means x cannot equal what?
yeah, so x cannot equal 1
Phew
(-inf, 1) U (1, inf)
Yep got it
Oh yes also the functions are f(-4) and f(3) for that question
Those aren't functions
Values
if there is more information though just send a picture please
Yes
so I can understand what you mean better
okay so
what it is saying is to evaluate the function at x=-4 and x=3
that is just another one of the things you need to do
Yes
Yes the domain
What do you think the domain is now?
(-inf, 1)
no, the domain of the function in your question
not the example I gave
this is a graph of your function
Wait so it has a arrow
Means it can’t be infinite
Right
Nvm its@oppisite
Arrow means infinite
So domain would be
(2,-inf)?
I’m confused about this graph ngl
No, stop for a second, I'll try to explain
Yes
your screen is not infinitely large right
so you can't see all of the graph
the arrow just indicates that it keeps going
even though you can't see all of it
because your screen is not big enough
True
so all the arrow means, is that the graph keeps going in that direction
even though the graph gets cut off on your screen
Now, as for the domain, remember what we are looking for.
Are there any x-values that you can not put into this function?
If there were, they would not have a cooresponding y-value on the graph
So what do you think is the domain?
Then it should be infinite
By infinite, do you mean that we can put all x-values into our function?
We can put all values of X in this, yeah
Yes I have better clarity now
I’m not looking at what numbers the arrows are on instead what all they can have values for
So it’s better
Okay great. The next part of your question asks for the range
Yes
the range is similar to the domain
so, now you know that the domain is all x-values you can put in
the range is all y-values you can get out
Got it
Oh wait so it’s 2 and above
Do you mean the range is 2 and above?
It doesn’t touch 1.5 or 1 or below but it touches 2 to be exact
True
So basically the graph is intercepting 2 and going above
not quite
Do we write infi in front of@the number
the open bracket means that you are excluding 2
it is [2, inf)
meaning that we are including 2
Ooo I see
and we write infinity after the 2 because it is larger than 2
So always [, to include
Got it
The next part asks about the x-intercepts
Yes
that is when the function touches the x-axis
does your function ever touch the x-axis?
yes
what about the y intercept
the y intercept is when your function touches the y-axis
does your function ever touch the y-axis?
so the y-intercept is 2
Yep
Also a question , there can’t be more than 1 y intercepts right?
Oh I see for range ?
yo can someone help me out with interval notation
increasing and decreasing intervals
this channel is occupied, read #❓how-to-get-help and open an available channel
Oh yea my bad i wrote that before but now I understand
okay just making sure 👍
yes it asks for f(-4) and f(3)
yes
this is saying, what is your function when x=-4, and what is your function when x=3
you should be able to tell by looking at your graph
Yep
great, all done then! good work
Wait so for -4 it doesn’t touch anything
For the value
Yep other than that we are done
Thank you so much @quaint prawn !
What do you mean? f(-4) is asking for the y-value of your function when x=-4
Yes got confused for a second, i got the values it’s correct :DD

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Can someone just explain the mapping, domain and range to me? I'm aware of what all three mean but just in this context because I missed the lesson these notes are from.
Like why is it X+2 for the mapping and not X-2?
And how they got the domain and range
are you sure those answers are right
its what the teacher did for notes
are those circles on the original curve hollow or filled in
it looks hollow?
idk if thats important tho
i just wanna know why (x-2) makes it move 2 units right and not 2 units left?
its important because for the domain and range you used brackets, which implies a closed interval ( including end points)
i've never seen mapping used in precalc context, but i assume its because they want the translations to reproduce the old function from the new one?
the graph doesn't continue in the sketch so ig it is closed
if the circles are hollow it should be open interval, since those points aren't defined.
but its hard to tell from the picture
i just wanna know the whole x thing
does (x-2) not mean you're - 2 from the original x values?
if not
then does
x+2 would undo the translation of the transformed function
(x+2) make it translate left?
but shouldn't adding make the graph go right? and taking make the graph go left?
thats how the x axis works
g(x + 2) - 1 = f(x + 2 - 2) - 1 => f(x)
i think thats what they are trying to say, again, ive never seen "mapping" used in precalc, so just a guess.
i might as well ask the teacher lmao
i missed the lesson she went over these notes
i understood the earlier parts but this kinda fucked me up
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how to convert 1/3 to base 7 (by writing 1/3 as a sum of a geometric sequence)
Maybe use formula for geoemtric series limit.
@sly grove Has your question been resolved?
im not sure how to tho
the limit of a geometric series built with a is 1/(1-a)
you want limit 1/3
equate and solve for a (and hope it ends up being a digit)
can you give an example? (also i haven't learned limit of geometric series
$1 + a + a^2 + a^3 + a^4 + \dots = \frac{1}{1 - a}$
M8732
You have learnt geometric series but not this?
that's right
odd
but it basically just does what it says.
so we want
I think I said it incorrectly above.
We do not want to solve for a, as a = 1/7
is there a domain of values for a?
yes, it only converges between -1 and 1
,wolf 1/(1-1/7)
So if we had the number
1.1111111111111111111111111111111111... base 7
it would be 1.1666666... or 7/6
Let's work from here.
also, there's an example problem (if you want to see it
I would prefer if we focus on the problem first.
its solution may be reused on this problem (but i dont know how)
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✅
@sly grove Did you understand my sum?
this one? 1.1111... = 1 + 1/7 + 1/7^2 .... = 1.1666...?
1/(6/7) = 7/6
yup
Just making sure
wpe are usually not even supposed to give away solutions 
Anyway, now we got 7/6
in base 7
How can we get from 7/6 to 1/3 using operations
that do not mess up the base-7 representation in unpredictable ways?
7/6 = 1 + 1/6, and then get from 1/6 to 1/3?
yeah. how would you go from there to base 7 then?
Well we know
7/6 = 1.1111111111111111111111111111...
we subtract 1
and then we multiply by 2
Both are not rocket science for either fractions nor base-7
(1.11111...(7) - 1) *2?
yes
Easy, right?
thank you so much:))
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i got to 11x = x(a+b) + 2(a+b)
im not sure where to go from here
a + b = 11
but what about the second part
2A + 2B = 0 maybe?
idk 😨
how would you do it if it was 11x + 1?
so how would you do it if it was 11x + 0
Yes
yay!!
it is
yep
and so (-b) + b = 11? so 11 !=0 tho
does x^2 +4x + 4 not factor to (x+2)(x+2)
oh no
You will write
woops
11x/(x^2 + 4x + 4) = A/(x+2) + B/(x+2)²
Might help to use actual numbers if you want the intuition. PFD is basically:
5/6 = 5/(2×3) = A/2 + B/3 = 1/2 + 1/3
also in that case would it be 11x = A/(x+2) + (Bx + C)/(x+2)²
okok
huh?
11x/(x^2+4x+4) actually
oh yeah
Until you clear the denominator
thats what i meant sorry
isnt that only for when a quadratic factor cant be broken up into smaller factors (aka it doesnt have real zeros)
Repeated roots are special
yeah but if they're real theres no need for Bx + C in the nominator
There is. Because the denominator is of degree 2
So the numerator can be of degree 1
Hopefully that makes sense q
so what i got after separting everything was
a + b = 0
4a + 2b + c + 4 = 11
2c = 0
so c = 0
4a + 2b + 4 = 11
so 4a + 2b = 7
a = -b
-4b + 2b = 7
im pretty sure thats wrong
b = -7/2
the case you're talking about is for when there are imaginary roots
with the real ones, A/(x+2) + B/(x+2)^2 is enough
i got it wrong 💀
ok sorry lol
nahh no worries, nothing to apologize for
anyway this is what you're supposed to have now
ahhhh i messing up somewhere even tho im using this one
so i have like 11x = A(x+2)^2 + B(x+2) right
so i expand to 11x = (x^2)(A) + x(4A + B) + (4A + 2B)
wouldnt that mean A = 0 and B = 11 ?
wait wait how did you get this
oh wait lol
from here, try multiplying both sides by (x + 2)^2
dont rush, thats how most of those types of mistakes are made 😭
almost
11x = a(x+2) + b
yep, you got it :D
thanks 😭
you're welcome!
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HOw in the world is this wrong??
show your work maybe?
Ok
7/4 + 2/4 is not 11/4
very true
7/4 + 8/4 is also not 11/4
it is b^(1/2) yes
ah
thats the mistake i mde then
so it will be so then 1/2 + 5/4
get common denom
so will be 7/4?
where did 5/4 come from?
make sure you have the right parentheses, but yes
Oh ok
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why can -x^2 + x + 6 simplify to (x-3)(x+2) but when its in a function y = -x^2 + x + 6 you cant
you can
Why not
show screenshot or something
ask it to factor instead
💀 oh you right lmfao
"simplification" is somewhat nebulous
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I’m stuck at b. I found the limf(x) by doing g(x)=xf(x)-2x+3/x+2 etc etc but I can’t use this at b
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Hello, why are these two formulas for combinations without repetition equal
can someone show me an example like when n = 5 k = 3
@torn jolt Has your question been resolved?
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i dont speak very good spanish
sorry
mm you don't need to, it's just the formulas, but I will translate
Non-repeating combinations
The number of combinations of n elements taken from k and k as well as groupings of the same number of elements is different, at least in one element, is calculated with the following formula:
why are these two formulas for combinations without repetition equal
can someone show me an example like when n = 5 k = 3
<@&286206848099549185>
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A father has his wedding anniversary and birthday on the same day.
The family celebrates with a trip to London. The children pay for the musical and football match for the father. The mother pays the ticket for the football match itself, but gets a ticket for musical by the children. Tickets for the musical cost SEK 2,500 in total, while the tickets for the football match cost SEK 5,400 in total.
together. The plane tickets cost SEK 1,600 per person.
Write an expression for how much each of the children needs to pay.
I might struggle with converting my expression to LaTex format, but here we go:
The answer, according to the book, is $2500/x + 5400/x+1 + 1600$
Seed
Do $\frac{a}{b}$
ItzKraken
x = amount of children btw
The answer, according to the book is:
$\frac{2500}{x} + \frac{5400}{x+1} + 1600$
grrr
Don't put multiple $
right
Only one $ at the start, one at the end
Seed
Perfect, thanks @elfin stream !
Anyways, I disagree with the equation
I believe it rather should be $\frac{2500}{x} + \frac{5400-\frac{5400}{x+2}}{x+1} + 1600$
Seed
Is the father going to the musical 
yes, atleast in this problem
Hmm and so is the mother?
yes, and she is getting her musical ticket paid by the children
everyone will pay for the 1600 SEK flight ticket,
the 2500 SEK is the cost for everyone to get into the musical. This will be split between the children
5400 is the total cost of the football match. Children split the bill for the tickets that it costs for them and the father, while the mother pays for herself
if we were to analyse my expression above, we would see that 2500/x is logical, considering that 2500 is the bill for the entire family, which will get split by the amount of children, x
Yes
the +1600 also is logical, since everyone pays for their own flight ticket
1600 is per person
Hmm the books answer is correct
this expression?
What does 5400-(5400/(x+2)) represent? Total - the mother's payment right?
Where is it written she pays for herself
yes, we know that 5400 SEK is the total cost for entire family
Oh found it
and we want to make it so that the children split the bill between the ticket cost for them and the father only
Yeah
we need to therefore remove the cost of the mothers ticket from the 5400, and split the difference number that we get between the children
how do we do that? we know that x is the amount of children, and that there is both a father and a mother. This means that 5400/(x-2) tells the cost for each ticket
one ticket costs in other words 5400/(x-2)
5400/(x+2) no?
and the mother pays for herself, so the amount that the children will split between themselves is 5400 minus 5400(x+2), which is the cost for the mother's ticket
The book however says it's just $\frac{5400}{x+1}$ and calls it a day
Seed
They must be then assuming 5400 SEK doesn't include the mothers share
@elfin stream it might very well be the book's perspective, in which i comment that the problem formatting was vague
because seconds after, another problem appears where it's told that the children pay a total of 3000 SEK, so you're supposed to make an equation
using my method you get no whole x-values
using the book's method you get 5
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prove that for a,b,x positive integers $\gcd(x^a-1,x^b-1)=x^{\gcd{a,b}}-1$
bigpufik
im trying to use euclidean algorithim
im getting
$\gcd(x^a-1,x^b-1)=\gcd(x^a-1,x^a-x^b)$
bigpufik
i can assume a>b
and try to factor x^b
then
$\gcd(x^a-1,x^b-1)=\gcd(x^a-1,x^a-x^b)=\gcd(x^a-1,x^{a-b}-1)$
bigpufik
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how to solve $$\lim x -> 0 ( x floor(\frac{1}{x}))$$
neither
i know the so,ution is 1
but idk how to get there
suppose to use basic calculus limit properties
you can try to give bounds to floor(1/x) if you've seen the squeeze theorem
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'
I don't think this is right
Well I know that it's "right"
according to the book
But I don't think os
It doesn't make sense
Ping if you answer please
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@late fable Has your question been resolved?
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<@&286206848099549185>
what do u need
Look at the question
@late fable Has your question been resolved?
Yes
I know the answer is right but It doesn’t seem right
but when they say "Smahed into" yes responses are higher
Basically it's saying that if you use more violent words
That choice doesn’t make sense
they are more likely to respond yes
The treatment being the word choice
when you chose "smahed into" they were more likely to respond yes
It is tho
yea but we still predict no
Like no matter what word choice
We will always predict no
wdym
why’d you say ?
I think you're confusing most likely response with prediction
What
I’m not
I think you are
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Hello, how do I get the 1st term of an arithmetic sequence when I'm given only with the 3rd and 17th term?
I've tried using the arithmetic equation but I keep on getting wrong/invalid answers
at a loss here right now
Can you show your work?
,rotate
