strange crown
I get why the equation on the left is a quadratic exp equation( blue text), but curious as to why the equation on the right(red text) is not.
My guess Is it’s because the two terms are not of the same base? So technically they don’t follow the same power. So solving for one of the x terms doesn’t solve for the other one?
pallid halo
yea, in the first one you could let y = 3^x and you get a quadratic in y
this won't work in the second one because as you noted, the bases are not the same
also, even if the bases were the same, you have a 9th power term, so if anything it would be 9th degree polynomial, not a quadratic
strange crown
also, even if the bases were the same, you have a 9th power term, so if anything...
Ahh that makes sense, so it’s a 1st degree polynomial if you leave it at base of 8 but at base 2 it becomes 9th degree. But since the two terms aren’t/can’t be converted to same base unless you use log func, can’t just factor like a normal quadratic
Anyway I think that clears that up for me, thanks
.close
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tall moon
simplify $\frac{\sin(7x)-\sin(5x)-\sin(3x)+\sin(x)}{\cos(7x)-\cos(5x)+\cos(3x)-\cos(x)}$
potent lotusBOT
skissue.in.a.teacup
tall moon
is this correct?
flat gorge
,w [sin 7x - sin 5x - sin 3x + sin x]/[cos 7x - cos 5x + cos 3x - cos x]
potent lotusBOT
flat gorge
tan 2x it is 
tall moon
thank you!
astral mural
You have good eyes
tall moon
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tall moon
i have my reasons
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slim cove
im trying to find the relative extrema and points of inflection
but i think myrelative extrema is wrong
and also idk how to find the second dereviative
unique jackal
and also idk how to find the second dereviative
the 2nd derivative is a little bit of a pain to find, but you can use the product rule
crystal forge
before doing derivatives you seem to be simplifying the e's exponent and you wrote -1/2 x^2 + 2x - 2 = -x + 2, kind of confused what you are doing there
oh derivative of that part i see
it'd be helpful to actually write what you are doing instead of just a bunch of = signs 😄 between expressions that aren't actually equal
calm coralBOT
surreal patio
btw shouldnt the sign chart be + and then -?
unique jackal
hmm ok ill try
as soosh said, it would be really helpful if you just indicated you were simplify and differentiating the exponent here 
surreal patio
cuz if u choose a number bigger than 2 , then the entire thing becomes negative :d
surreal patio
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surreal patio
ur not doing it with g'(x) :c
surreal patio
u even wrote it there , signs of f'
slim cove
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tall moon
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tall moon
so,
3ab=a+b => 3abc=ac+bc
4bc=b+c => 4abc=ab+ac
5ac=a+c => 5abc=ab+bc
12abc=2(ab+ac+bc)
24abc=4(ab+ac+bc)
24abc/(ab+ac+bc)=4(ab+ac+bc)/(ab+ac+bc)=4
is this correct
ok ty
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tall moon
how many triangular pyramids can be formed by using the verticies of a cube
tall moon
total is 8c4=70
but there are some that are just fully flat so not a pyramid
so there are the ones that are like the faces of a cube, so subtract 6
then the diagonals that are horizontal to eachother, which is 2, and diagonals that are vertical, which is 4, so 70-6-6=68
did i miss anymore
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tall moon
bro i sent that like 5m ago it just close aa
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tawny coyote
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tawny coyote
can anyone help me with this limit
hollow totem
What's with the alpha?
this is so bizarre
First try to find the limit of the thing in the arctangent
sand sky
What's with the alpha?
Ig we have to split in cases and solve for different range of values of alpha
hollow totem
Ig we have to split in cases and solve for different range of values of alpha
well there can be at most one value of alpha for which the limit is finite and nonzero
sand sky
well there can be at most one value of alpha for which the limit is finite and n...
Yeahh !
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tawny coyote
..
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crystal marten
I gave an exam today and really wanna know if I passed ….. please help me confirm how much I scored … only the questions which I have ticked.. thanks in advance
swift dragon
!ss
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raven crest
Can someone please help me on this question
calm coralBOT
Can someone please help me on this question
raven crest
The red part is what I know so far
I don't know how to calculate the area of the second triangle
sand sky
Area Formula for scalene triangles
raven crest
yeah i noticed that too
raven crest
Area Formula for scalene triangles
i don't know what the height is
sand sky
i don't know what the height is
Ofc 5, if 12 is base
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lament prawn
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next nest
How do I find what this limit converges to?
calm coralBOT
How do I find what this limit converges to?
next nest
Tried using squeeze theorem but I don’t know what lower bound to use
eternal shard
what's x_1 and x_2
glad parrot
Why x,y became x1,x2
tidal matrix
isnt the lower bound just 0
flat gorge
Honestly, step one should be substituting the x_1 - 1 = X
eternal shard
ln(x) can become negative
next nest
lnx1 isn’t always positive
winter herald
Why do you write it is 1?
next nest
It’s a question mark
flat gorge
$\lim_{(X, Y) \to (0, 0)} \frac{XY^2}{X^2 + Y^2} \cancelto{1}{\frac{\ln (1+X)}{X}}$
winter herald
wow
tidal matrix
near x = 1, if x > 1, lnx>0
if x < 1, lnx < 0
as x -> 1, lnx -> 0
therefore we know |lnx| is small for x near 1
potent lotusBOT
Arya
next nest
Is that the same function?
flat gorge
Is it not? Is it easier to deal with (x_1 - 1)² + x_2² and ln(x_1) as (x_1, x_2) -> (1, 0) than to deal with X² + Y² and ln(1+X) as (X, Y) -> (0, 0)?
tidal matrix
or you can use squeeze theorem, you can see that as x -> 1, lnx -> 0 therefore lower bound is 0
next nest
Yeah or I wasn’t sure about the lower bound
tidal matrix
yes lower bound
if x tends to 1, lnx will tend to 0
so you can consider the lower bound to be 0
winter herald
How does that help
tidal matrix
because the definition of squeeze theorem is g(x) <= f(x) <= h(x)
next nest
Yeah and both upper and lower tend to 0 then
tidal matrix
Yeah and both upper and lower tend to 0 then
yes
winter herald
**Arya**
I think somehow a minus got lost here?This is now >= 0.
There is no lower bound
tidal matrix
as x->1, lnx->0
tidal matrix
~~It is negative~~
where?
flat gorge
I think somehow a minus got lost here?This is now >= 0.
X -> 0^{-} => ln (1 + X) < 0
flat gorge
x_2
tidal matrix
your x2
winter herald
,wolf lim ((1/n)²ln(1-1/n))/((1-1/n-1)²+(1/n)²), n->oo
potent lotusBOT
winter herald
huh wait
mortal orbit
the limit is indeed zero, and there is a way to prove this term goes to 0
rewrite it as
$\frac{xx_2^2}{x^2+x_2^2}$
potent lotusBOT
rafilou is not not born in 2003
mortal orbit
well it does
winter herald
choose x_1=1 and x_2=1/n
next nest
How do I use Cauchy schwarz to show that?
mortal orbit
~~choose x_1=1 and x_2=1/n~~
then f(n) = 0 for all n
next nest
abs(xy)<=abs(x) abs(y) is Cauchy schwarz right
mortal orbit
abs(xy)<=abs(x) abs(y) is Cauchy schwarz right
no
wait
I might have gotten confused on the name
it's AM GM
next nest
What’s that
next nest
Or maybe using polar coordinates?
potent lotusBOT
rafilou is not not born in 2003
next nest
mortal orbit
deduce an upper bound of $|xx_2|$
potent lotusBOT
rafilou is not not born in 2003
next nest
x x2 + 1 maybe?
mortal orbit
maybe rather consider $(|x|-|x_2|)^2$ instead
potent lotusBOT
rafilou is not not born in 2003
mortal orbit
x x2 + 1 maybe?
and no it won't come like that
mortal orbit
**rafilou is not not born in 2003**
you need to rewrite what $(|x|-|x_2|)^2\geq 0$ means
potent lotusBOT
rafilou is not not born in 2003
next nest
Yeah
next nest
Doesn’t it matter that the denominator is 0
flat gorge
We’re trying tryna show x x2/(x^2 + x2^2) converges to 0
That does not converge to any limit, the original expression does.
next nest
So like squeeze theorem?
mortal orbit
multiplying by x_2, that goes to 0
yes it's the idea
so we're trying to find an upper bound
next nest
That does not converge to any limit, the original expression does.
Yeah but since the ln term becomes 1 we wanna show this part goes to 0
mortal orbit
Yeah but since the ln term becomes 1 we wanna show this part goes to 0
they're saying that because you forgot a ^2 in your message
you wrote xx_2/... instead of xx_2^2
anyways
mortal orbit
**rafilou is not not born in 2003**
rewrite this in a way
mortal orbit
**rafilou is not not born in 2003**
that makes this term appear
flat gorge
**Arya**
Coming back to this, use the inequality to find appropriate upper bound to this
mortal orbit
And not a lower?
what's the lower bound of the absolute value of something?
next nest
0
mortal orbit
so we only need an upper bound
mortal orbit
**rafilou is not not born in 2003**
for the absolute value of this
next nest
Where does the absolute term come
mortal orbit
wdym
next nest
Or I’m confused why were looking at the absolute value now
mortal orbit
we want to show our thing converges to 0
next nest
So we show absolute convergence?
mortal orbit
so by squeeze theorem
next nest
Is that the term I don’t remember
mortal orbit
no that's not it
it's just that
f(x,y) -> 0 iff |f(x,y)| -> 0
so trying to show |f(x,y)| -> 0
upper bound |f(x,y)| by something that goes to 0
and we're done
next nest
Okay so then we look for an upper bound to the absolute value of the expression x x^2/(x^2+x2^2)
So for example taking the x^2 term away from the denominator were left with just x which goes to 0
Or?
How? The term becomes bigger
Since were dividing by a smaller number
flat gorge
$\frac{X}{1 + \frac{X^2}{Y^2}} \leq X$
potent lotusBOT
Arya
next nest
Oh that’s much simpler
flat gorge
And since the RHS converges to 0, by squeeze theorem you have your result
flat gorge
in abs value
Exactly 
mortal orbit
otherwise
next nest
Isn’t this result tthe same as taking away the x^2 term from the denominator
Of the absolute value
mortal orbit
Of the absolute value
yes it's the same as upper bounding the denominator x^2 + y^2 >= y^2
flat gorge
Put abs on | • |
mortal orbit
Or forgot the absolute sign
and that
next nest
Like this or
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exotic cosmos
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exotic cosmos
i have to show that the number of incorrect answers by ahmed < 4
idk if i’m making this problem harder than it is but:
x is the number of questions he got right in section 1 and 30 - x is the number of questions he got wrong in section 1
The same thing applies for y and z but in sections 2 and sections 3 respectively
And so:
x(1) + (30-x) (-3) + 2y + (30 - y)(-1) + 2z + (15 - z) (-2) = 4x + 3y + 4z - 150
now i know that 0 < x,y <= 30 and 0 < z <= 15, where x,y, and z are integers
i'm guessing then the problem reduces to me having to show that 75 - (x + y + z) < 4 or in other words (x + y + z) > 71 when 4x + 3y + 4z > 258
looks pretty hard but with code i’ve verified that the claim is true
calm coralBOT
@exotic cosmos Has your question been resolved?
calm coralBOT
@exotic cosmos Has your question been resolved?
exotic cosmos
<@&286206848099549185>
exotic cosmos
i already did all that tho lol
check my message above
there’s 3 variables
for 3 sections
exotic cosmos
i have to show that the number of incorrect answers by ahmed < 4
the question is literally this tho
if your solution is something along the lines of having 4 mistakes in section 2 still not resulting in <90% grade thus …
then that feels gimmicky
yeah then that isn’t 😭a type of solution i’m looking for
it’s gimmicky
can u build off what i did
no, i thought of the exact same solution
which is why i typed it out before u did
it works for this problem
i want to learn smth from the problem
instead of just “solving” it
exotic cosmos
isn’t this 😭just what we said informally
can we just do what i did
at the start?
flat gorge
Did you try case solving this? The four cases being Ahmed's scores at 82, 83, 86, 90
flat gorge
i have to show that the number of incorrect answers by ahmed < 4
Wait that's all?
exotic cosmos
if your solution is something along the lines of having 4 mistakes in section 2 ...
not something like this solution
flat gorge
Wdym no gimmicks
exotic cosmos
bc that solution rests more on the number 4 than anything and therefore i’m not learning anything from this problem
flat gorge
Welp, that's the way though. An incorrect answer in sections 1 and 3 give a minus 4 score
exotic cosmos
other than “solving” it
exotic cosmos
Welp, that's the way though. An incorrect answer in sections 1 and 3 give a minu...
😭😭
flat gorge
In section 2 is a minus 3
exotic cosmos
the thing is
the question type is of the form
is 4 > incorrect answers from ahmed
4 = incorrect answers from ahmed
4 < incorrect answers from ahmed
no sufficient information
flat gorge
Do you want an optimization type solution?
flat gorge
Yes. In fact 3 > incorrect answers from ahmed
flat gorge
Write all restrictions on x, y, z, even something like x ≥ 0 or x ≥ 1
exotic cosmos
isn't it already written there
flat gorge
Oh hmm
exotic cosmos
tbh
our problem is now just this:
like all the yap probably only requires this
(according to me)
but i think all steps logically follow
flat gorge
So, the problem is to minimize (x + y + z) subject to 4x + 3y + 4z > 258, 0 < x, y ≤ 30, 0 < z ≤ 15
exotic cosmos
that could be another way to phrase the problem yeah
actually show that the minimum > 71
exotic cosmos
ran a code with this description
exotic cosmos
and as you can see yeah the sum > 71
exotic cosmos
hmm okay np.
not smth overly complicated either 😭
this is assumedly algebra
bitter maple
> x,y, and z are integers
> 0 < x,y <= 30 and 0 < z <= 15``` surprisingly, that's technically all the problem boils down to^ imo
flat gorge
You might be looking for this
exotic cosmos
it's a 2 minute problem
i did think of linear programming
but like not too familiar with it
flat gorge
Is indeed a 2min prob if you're familiar with linear programming
flat gorge
it's a 2 minute problem
Give a sec.
> x, y, and z are integers
> 0 < x, y <= 30 and 0 < z <= 15
=> 4(x + y + z) > 258 + y
=> x + y + z > 64.5 + y/4
Claim: y > 26
Assume to the contrary that y ≤ 26
=> 4x + 3y + 4z ≤ 4(x + z) + 3*26 ≤ 180 + 78 = 258 =><=
Hence, y > 26
=> x + y + z > 64.5 + y/4 > 64.5 + 26/4 = 71
=> x + y + z > 71```@exotic cosmos that's probably the rigor you're looking for?
I will also check Linear Programming and provide an alternate soln. once I'm done revising :O [been 2 yrs since I've touched the topic]
calm coralBOT
@exotic cosmos Has your question been resolved?
still glade
A question for a game (Factorio)
calm coralBOT
A question for a game (Factorio)
still glade
I need to produce 16 (A) items every 21 seconds, they need a material that takes 15s to craft, how many crafters do I need at the same time to supply the (A) items
i feel stupid ngl
glass heart
the key to these questions is to make the timeframe the same
for example by considering the lcm
(least common multiple)
still glade
thanks for clarifying
glass heart
what is the lcm of 21 and 15?
still glade
3?
glass heart
thats the gcd, greatest common divisor
still glade
what is the lcm of 21 and 15?
105?
glass heart
yes
so, how many items do you need to produce in 105 seconds?
how many materials do you need?
how many do you make from one crafter in 105 seconds?
glass heart
not quite. you probably mean the correct thing tho
you said you need to produce 16 items in 21 seconds, so thats how many in 105 ?
glass heart
if you need 16 in 21 seconds, then you need 80=5*16 in 105 seconds
still glade
no like I meant I produce 5 every 105
glass heart
ok we are clearly talking about different things
I can only go by your initial numbers
calm coralBOT
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mortal oyster
how would we show f is into
calm coralBOT
how would we show f is into
mortal oyster
oh wait is it that f is always positive in the given domain
tidal grotto
it's quicker to note f(x) is an even function ;3
note the numerator is x^2-4 as well, and this has two zeros that are in the domain of the function
hollow totem
wtf is an into function 
tidal grotto
injective
hollow totem
and many one into?
tidal grotto
not injective
tidal grotto
oh wait
hollow totem
one-one onto means injective and surjective?
and many one into means not injective?
tidal grotto
im tired
hollow totem
that is absurd notation
hollow totem
one-one into means injective
into alone also means injective?
mortal oyster
onto is surjective
hollow totem
that one i know
mortal oyster
one one is injective
mortal oyster
it's quicker to note f(x) is an even function ;3
that just proves injectivity tho?
tidal grotto
yes
mortal oyster
.close
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noble crown
$\sum_{i=1}^{n} i^4$
calm coralBOT
$\sum_{i=1}^{n} i^4$
potent lotusBOT
kronium_
noble crown
$$ Is it possible to find the Sum of this using the telescoping series?
$\text$ Is it possible to find the Sum of this using the telescoping series?
potent lotusBOT
kronium_
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
ancient thistle
yes
noble crown
How
ancient thistle
,w expand 1/5 (x + 1/2)^5 - 1/5 (x - 1/2)^5 - 1/6 (x + 1/2)^3 + 1/6 (x - 1/2)^3 + 7/240
noble crown
$U_i = i^4, U_{i-1} = (i-1)^4$
noble crown
,w integrate x
potent lotusBOT
noble crown
,w integrate Li_2(x)
potent lotusBOT
noble crown
,w integrate li(x)
glad parrot
?
potent lotusBOT
ancient thistle
,, U_i = \f 15 \parens [\bigg] {i + \f12}^5 - \f 16 \parens [\bigg] {i + \f12}^3 + \f 7 {240} \parens [\bigg] {i + \f12}
use this
noble crown
u literally explained nothing
but thanks for trying
answer isn't what I need
its how I get to it
potent lotusBOT
ancient thistle
well you can construct it
noble crown
construct?
ancient thistle
just go term by term
like
look at the first term
,w expand 1/5 (i + 1/2)^5 - 1/5 (i - 1/2)^5
potent lotusBOT
noble crown
yeah but how'd u get that
ancient thistle
,w -1/6 (i + 1/2)^3 + 1/6 (i - 1/2)^3
noble crown
,w expand 1/5 (i + 1/2)^5 - 1/5 (i - 1/2)^5
how'd u get that function
potent lotusBOT
ancient thistle
how'd u get that function
well you need i^4
noble crown
yes
ancient thistle
so
ancient thistle
god wolfram is using the imaginary unit
celest hearth
Hello is this occupied?
noble crown
bruh yes its occupied
glad parrot
Hello is this occupied?
Yes
potent lotusBOT
ancient thistle
so you see here
noble crown
,w expand (k+1)^5
potent lotusBOT
ancient thistle
when you take a difference of 5th powers you get 5k^4
noble crown
u have a lot of things u dont need here
ancient thistle
yes
which is why i didn't do +1
i did +1/2 -1/2
it makes a lot of the terms go away
you re-centre on 0
you have to divide by 5 to get rid of the 5 coefficient on the k^4
noble crown
,w expand (k+0.5)^5
potent lotusBOT
ancient thistle
,w expand (k+1/2)^5 - (k-1/2)^5
potent lotusBOT
ancient thistle
see how the difference has less extra stuff
flat gorge
We telescope $(n + 1)^5 - n^5 = 5n^4 + 10n^3 + 10n^2 + 5n + 1$ to figure summation $n^4$, given summation $n^3, n^2, n$. So you could plug the rest and get your desired closed form for sum $n^4$. But yes, $(n+1/2)^5 - (n-1/2)^5$ is smart
ancient thistle
so then you construct more terms
noble crown
5k²/2
potent lotusBOT
Arya
noble crown
wait how bout this
ancient thistle
,w expand 1/5((k+1/2)^5 - (k-1/2)^5)
potent lotusBOT
noble crown
,w expand (k+1)^5 - (k-1)^5
ancient thistle
we need to rid the k^2/2 term
potent lotusBOT
noble crown
THATS A LOT CLEANER
ancient thistle
then it's not quite telescoping but okay that's fine
you get difference of 2 between terms
noble crown
yeah but u still get it right
ancient thistle
you don't get direct cancellation between adjacent terms
yeah
sure
we can work with that
so we should do 1/10 of that
noble crown
you don't get direct cancellation between adjacent terms
why not, and why does the other one have direct cancelation
ancient thistle
,w expand 1/10((k+1)^5 - (k-1)^5)
potent lotusBOT
noble crown
because k^4 doesn't have a coefficent?
ancient thistle
no
because when you have a telescoping series it's like
,, \sum (u_k - u_{k-1})
potent lotusBOT
ancient thistle
so the terms are like [ u_1 - u_0 + u_2 - u_1 + u_3 - u_2 + \dotsb ]
potent lotusBOT
ancient thistle
and here you get direct cancellation between adjacent terms
noble crown
ik that
but how do you recognize the direct cancellation
like the function that you gave
ancient thistle
well
ancient thistle
i'm suggesting to do like
noble crown
i'm suggesting to do like
u'll suggest a course 😭
ancient thistle
(k + 1/2)^2 - (k - 1/2)^2
k + 1/2 and k - 1/2 differ by 1
k + 1 and k - 1 differ by 2
noble crown
yes
ancient thistle
when you differ by 1 is when telescoping happens
ancient thistle
i mean you can still telescope when it's a difference of 2
it's just not as nice
like it's doable
noble crown
$\sum_{n=1}^{2025} n(n+1)(n+2)(n+3)(n+4)(n+5)$
ancient thistle
okay but like again this will shift only by 1
ancient thistle
actually no will it
noble crown
here
ancient thistle
,, \sum_{n=1}^{2025} n(n+1)(n+2)(n+3)(n+4)(n+5)
potent lotusBOT
ancient thistle
lowercase s
potent lotusBOT
kronium_
noble crown
yes
ancient thistle
okay so this is like
ancient thistle
a long one
ancient thistle
you need to write out a hell of a lotta terms
ancient thistle
6?
noble crown
when u expand
ancient thistle
but doable
ancient thistle
well
ancient thistle
only after 6 terms
noble crown
no
after the first time
actually
from the second term onwards
u get cancellation
ancient thistle
are you sure
noble crown
why
ancient thistle
okay i think it might work
noble crown
look
ancient thistle
i'm just checking it in my head
ancient thistle
yes
ancient thistle
okay
ancient thistle
these numbers will get large quick
sinful field
Hi I am new
noble crown
wait no i messed up
ancient thistle
hmm
ancient thistle
sure
noble crown
ok so
ancient thistle
,w expand C(n+7,7) - C(n+6,7)
potent lotusBOT
ancient thistle
wolfram is so bad
noble crown
I THOUGHT'D WE'D GET CANCELLATION
ancient thistle
no you have to like
write it differently
this is annoying to type on a phone
,w simplify (n + 7)(n + 6)(n + 5)(n + 4)(n + 3)(n + 2)(n + 1) - (n + 6)(n + 5)(n + 4)(n + 3)(n + 2)(n + 1)n
noble crown
wait
ancient thistle
okay this is just
potent lotusBOT
ancient thistle
jesus
there
you do that thing
so it's a shift by 1
if you like
do the algebra properly
ancient thistle
yes
noble crown
(x+6) - (x-1) = 7
ancient thistle
and you take the difference of those two
noble crown
the function multiplied by (x+6) is called p(x)
ancient thistle
and you get your function
noble crown
the function we want is p(x) - q(x)
ancient thistle
going from p to q
noble crown
and q(x) = p(x-1)
noble crown
SEE
ancient thistle
but because of the shift by 1
noble crown
so theres cancellatiion
noble crown
but because of the shift by 1
by 7
ancient thistle
and q(x) = p(x-1)
you even wrote it here
noble crown
YES BUT HOW DO U KNOW THE SHIFT
noble crown
without evaluating the TERMS
ancient thistle
well
noble crown
you wrote it
I was told the answer
ancient thistle
the factors of p are like
x up to (x+7)
and the factors of q are (x-1) up to (x+6)
so you can see the factors of q are the factors of p shifted down by 1
that's why you get the equation q(x) = p(x - 1)
and my point before is that it's the same for (k + 1/2) and (k - 1/2)
you shift by 1
ancient thistle
yes
noble crown
,w expand (k+1)^5 - (k-1)^5
potent lotusBOT
ancient thistle
okay now the 5th powers cancel
ancient thistle
sure
noble crown
,w expand (k+ 1/2)^5 - (k - 1/2)^5
potent lotusBOT
ancient thistle
0?
ancient thistle
where about
noble crown
1k^4
noble crown
yes
ancient thistle
,w expand 1/5((k+ 1/2)^5 - (k - 1/2)^5)
potent lotusBOT
ancient thistle
the 5 comes from the fact that we used 5th powers
ancient thistle
and binomial theorem
ancient thistle
okay speak
ancient thistle
we will keep putting on terms
to cancel them
so for example
,w expand (k+1/2)^3 - (k-1/2)^3
potent lotusBOT
ancient thistle
look
we get a k^2 here right
so if we subtract off the right amount of this from the previous terms
we can get rid of the k^2/2
noble crown
yes
ancient thistle
,w expand 1/5((k+ 1/2)^5 - (k - 1/2)^5) - 1/6((k + 1/2)^3 - (k - 1/2)^3)
noble crown
wait
potent lotusBOT
ancient thistle
okay waiting
ancient thistle
ok
noble crown
$\sum_{i=1}^{n} i^4 = \frac{1}{5} \sum_{i=1}^{n} (i+\frac{1}{2})^5 - (i-\frac{1}{5})^5 \quad - \frac{1}{2} \sum_{i=1}^{n} i^2 \quad - n/80$
ancient thistle
ayo
noble crown
and make another summation to evalutate that k²/2
and then subtract that from the original
and do the same for the constant
ancient thistle
,w expand 1/5((k+ 1/2)^5 - (k - 1/2)^5) - 1/6((k + 1/2)^3 - (k - 1/2)^3)
that's what this correction term is essentially doing
i think that's probably a 1/2 not a 1/10
noble crown
i think that's probably a 1/2 not a 1/10
well
we are dividing the original equation by a 5 along with that right
cuz
its
noble crown
(1/5) (k + 0.5)^5 - (k-0.5)^5
ancient thistle
it's already divided
noble crown
no
I kept the constants different
the first 1/5 is only for that specific sum
not distributed
potent lotusBOT
ancient thistle
agreed?
potent lotusBOT
kronium_
ancient thistle
if you want to just sum the i^2 sum by itself you can
ancient thistle
the usual
noble crown
(i)(i+1)(2i+1)/6
noble crown
like
ancient thistle
maybe not?
noble crown
we are expected to memorize the quadratic equation but not the cubic equation
ancient thistle
its not like you'll spend forever rederiving it
noble crown
same way what about i^4
noble crown
k'
ancient thistle
but like
if you use it a couple times
i'm sure it'll stick in your head
can't be that hard to remember
noble crown
k
alright thx
i got maths exam tmrw
i gotta study for it
and this is not even taught in school this year
ancient thistle
ok
noble crown
hahaha 🤡
ancient thistle
ok
noble crown
the only thing I fear
ancient thistle
not just you
noble crown
u wanna see our question paper?
calm coralBOT
ancient thistle
noble crown
calm coralBOT
Available help channel!
sturdy edge
hello
sturdy edge
if AUB={1,2,3,4,5} and A∩B ={3} where A= {1,3,x} find the possible values of x in set A
x can be 2,4,5 right?
my ans key only shows 4 or 5
rustic osprey
dull wagon
um, the whole question seems inconsistent
sturdy edge
I meant intersection for the 2nd one
dull wagon
ah
sturdy edge
sorry
dull wagon
yeh, the values don't need to be ordered. its possible for x to be 2
sturdy edge
solution: since A Union B includes all elements 1,2,3,4,5 , the missing element must be covered by B and any choice of x. For A={1,3,x} , the values of x could be 4 or 5 for AUB to cover all required elements
that's the explanation
sturdy edge
i think you're correct
must be a typo then
silver tapir
I think they won't consider it bcz 3>2
that seems the only reason to include 4 and 5 and not 2
sturdy edge
but order doesn't matter ?
silver tapir
but order doesn't matter ?
ye ik
sturdy edge
ye ik
it's still typo anyways
the question should explictely state that order matters or something
earnest heath
if i have three vectors $$ l_i i cap, m_i j cap , n_i k cap, i = 1,2,3$$ they are all mutually perpendicular unit vectors
potent lotusBOT
lakshya
calm coralBOT
@earnest heath Has your question been resolved?
exotic cosmos
if i have three vectors $$ l_i i cap, m_i j cap , n_i k cap, i = 1,2,3$$ they a...
what
earnest heath
what
?
i cap j cap k cap
how u resemble vectors
in x y and z
axis
$$i^{}$$
potent lotusBOT
lakshya
earnest heath
$i^{\wedge}$
potent lotusBOT
lakshya
earnest heath
idk whateer
exotic cosmos
$\hat{\textbf{\j}}$
potent lotusBOT
mmmm7
exotic cosmos
i don't even understand your question lol
earnest heath
i don't even understand your question lol
prove that lines with direction ratios $$l_1,l_2,l_3; m_1,m_2,m_3 ; l_3,m_3,n_3 $$are also mutually perpendicular
dawn canopy
how do i prove
begin with what you already know, right down all the axioms you are aware of
earnest heath
also prove that those direction ratios are direction cosines...
dawn canopy
you could prove it by proving the angle between them to be 90deg
earnest heath
you could prove it by proving the angle between them to be 90deg
yes how to
dawn canopy
look up the formula for finding the angle
earnest heath
...
earnest heath
if you know the formula, it's clear it can't be done with the formula alone... anyway the formula is $$ cos(\theta) = l_1.l_2+m_1.m_2+n_1.n_2 $$ where $$l_1,l_2...$$ are direction cosines
potent lotusBOT
lakshya
dawn canopy
that's not the formula
earnest heath
yes it is
