help-0

[fading zephyr]

you need to do what ramonov and ann said

[lethal vortex]

can i please have an example hahahaha

kinda lost

[gray isle]

in the context of composition functions
the notation they used represents:
g( g(x))

[fading zephyr]

you have g(x) = ax + b

g(g(x)) means you substitute every x by g(x)

[lethal vortex]

a(ax + b)+ b

[fading zephyr]

mhm

[lethal vortex]

OH

I SEEE

LEMME TRY

[fading zephyr]

and now you have an order 1 term and an order 0 term

[lethal vortex]

so i need to expand and make it into (something)(something)

right

[fading zephyr]

no

[lethal vortex]

simplify*

[fading zephyr]

you expand it into a new polynomial of the form cx + d

which should be equal to 4x - 15

this will give you two equations

d= -15 and c = 4

but d and c are functions of a and b

so you solve for a and b from those 2 equations

[alpine sable]

||a(ax + b)+ b = 4x + 15||

[lethal vortex]

thanks!

[oak chasm]

@alpine sable Almost.

[lethal vortex]

im

[fading zephyr]

re

[lethal vortex]

edd

there is only an answer for that question right

[turbid idol]

discrete math any one knows the answer?

and this one too

[limpid mist]

Just see which of the intervals it holds for. Do you know how to read quantifiers?

[fading zephyr]

boxer, please try an empty channel

recon, no. you can immediately see that the a will be squared

meaning two values satisfy the equation in general

[lethal vortex]

i got

2 for a
and -5 for b

[fading zephyr]

what about the other value for a

[lethal vortex]

how is it possible to get another value

im sorry im not really bright

[fading zephyr]

a^2 = 4 and ab + b = -15 are the equations

there are two values of a for which a^2 = 4

[lethal vortex]

-a and a?

[fading zephyr]

mhm

so -2 is also a valid solution

and that means there's also another value of b that works

[lethal vortex]

so i can just add in a negative?

IM SO DUMB OMG

[fading zephyr]

i don't know what you mean by add in a negative

[lethal vortex]

uhmm

[ocean seal]

Chai T. Rex

[lethal vortex]

$a^2x + ba + b = 4x - 15$

[ocean seal]

recon

[oak chasm]

@lethal vortex So now you have two values for a. Solve b for the first value of a. Solve b for the second value of a.

[lethal vortex]

@oak chasm about this

if u have a variable squared

and if u want to move it

u need the

$+-$

[ocean seal]

recon

[lethal vortex]

thing?

[fading zephyr]

what is "move it"?

[lethal vortex]

is it always like this

[fading zephyr]

what chai is telling you is that there are 2 values of x

yes

[lethal vortex]

so the +- thing is neccesary

[fading zephyr]

yes

[lethal vortex]

thank you so much @fading zephyr and @oak chasm

cuz -- = +

[fading zephyr]

if you're seeing function composition, you have already learned thus

[lethal vortex]

i have

i forgotten

[errant dagger]

edddddddddddddddddd

hi anyone, if anyone has a good geometrical representation of a vector parametrisation of a circle please send it my way i would really love to have it thank you

[vale wigeon]

imagine a clock

[errant dagger]

ann

im imagining it

[vale wigeon]

imagine its hand spinning counterclockwise starting from pointing exactly to the right

[errant dagger]

yes

sure

[vale wigeon]

yeah thats your geometric representation

[errant dagger]

oh

hmm

maybe

im looking for

cos(t)i + sin(t)j

why that represents a circle of radius 1

[vale wigeon]

a circle of radius 1 centered at the origin

[errant dagger]

yes***

[vale wigeon]

try calculating the length of cos(t)i + sin(t)j

[errant dagger]

oh shit

1

[vale wigeon]

and observe that it is always equal to 1 no matter the value of t

[errant dagger]

oh waiwiaiwiwaiwi

and when u graph cos and sin

fucking with t

it describes all the points of a circle

i think

[manic glade]

hey need help with this
A,B matrices nxn
need to prove that if A^2 = AB and B^2= I+BA then A=0

[tired creek]

what have you tried?

[manic glade]

no clue where to begin with this honestly

i know i cant just say B=A because it isnt given that they are invertible

i dont see anything useful i can do with the given

i feel like it should be something i prove by contradiction

where i assume A is not 0

but as far as i know that gives me no info about B

[golden arch]

Can someone explain how to go from the second to the third step when calculating implicit derivatives?

[tired creek]

they differentiated again

[golden arch]

Yes but i dont understand where the last term comes from

When i calculate it i get 2 + 2y'² = 0

[tired creek]

product rule

[golden arch]

Aaaaah thanks!

[tired creek]

A^3=ABA=A(B^2-I)=AB^2-A

and with some manipulation from here you can end up with A^3=A^3-A@manic glade

[lethal vortex]

q29

uhm

my answer was

[ocean seal]

recon

[lethal vortex]

the answer sheet:

[ocean seal]

recon

[lethal vortex]

anyone know why?

[woeful pulsar]

^{-1}

use the diagram to help you calculate a and b

[lethal vortex]

oh

i see

thanks @woeful pulsar

[manic glade]

sick, thanks

[civic radish]

hihi!! could i please get help for question 10 a? im having a bit of trouble getting the answer please ping!!

[manic quail]

@civic radish , what exactly is the problem here? Just switch y and x, and then solve for y.

[civic radish]

do you mind looking at my working out?

[manic quail]

Not at all, just post it.

[civic radish]

in the answers, the bottom is a 6 but in mine i have a 3

[oak chasm]

,w solve x = 3y - 10 for y

[ocean seal]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by Chai T. Rex#7323

[civic radish]

ooooooooooo

[oak chasm]

Well, assuming x = 3y - 10 is a correct step, you're right.

[civic radish]

yayyyy

thank you so much!!

[oak chasm]

No problem.

[lapis harness]

hello how would one go about solving such DE

[ocean seal]

PBj4FPxrZF7ovZ

[lapis harness]

i assume you have to do a sub like this

[ocean seal]

PBj4FPxrZF7ovZ

[lapis harness]

but i'm lost after that

[lapis harness]

i dunno where to proceed from there it gets messy or i'm not seeing smth

nvm in retarded, this made everything clear

[vapid smelt]

Simplify

,tex $\frac{x^2}{x-y}$ + $\frac{y^2}{y-x}$

[ocean seal]

Inheritanc-e ♦

[vapid smelt]

How u do this?

[knotty sleet]

$=\frac{x^2}{x-y}-\frac{y^2}{x-y}$

[ocean seal]

Wheeler

[dapper umbra]

take an LCM lol

[knotty sleet]

$=\frac{x^2-y^2}{x-y}$

[ocean seal]

Wheeler

[knotty sleet]

=x+y

[ocean seal]

PBj4FPxrZF7ovZ

[hard thorn]

i have a question if tan(pi/12)=2-sqrt(3)

calculate tan 5pi/12

[glass lichen]

5pi/12 = pi/12 + 4pi/12

[hard thorn]

that won’t help cuz i. dont have pi/12

[glass lichen]

yes you do...?

[hard thorn]

i mean 4pi/12

i tried this

[glass lichen]

4pi/12 is an exact value

,rotate

[glass lichen]

4pi/12 = pi/3

[hard thorn]

tan(pi/12+pi/3)

equals to ?

[glass lichen]

compound angle for tan

$\tan(A+B)=\frac{\tan(A)+\tan(B)}{1-\tan(A)\tan(B)}$

[ocean seal]

moshill1

[hard thorn]

umm we didn’t learn this one

i got it thx

,rotate

[alpine sable]

The sketch below shows triangle ABC and triangle DEF. The two triangles are equal.
The length ratio between the two triangles is 1: 3.

How big is the area of ​​triangle DEF if the area of ​​triangle ABC is 10?

[lapis harness]

in 1d you scale by k

so in 2d you will scale by k^2

[ionic jewel]

yes but thats not so intuitive

better he understands why

[lapis harness]

you can always do

[ocean seal]

PBj4FPxrZF7ovZ

PBj4FPxrZF7ovZ

[alpine sable]

?

[lapis harness]

area of the first triangle is A

2nd one is A`

[alpine sable]

A^2

I'm confused what you mean

10^2 = 100

@lapis harness right?

and volume 10^3

[lapis harness]

nvm im blind

[alpine sable]

???

[lapis harness]

if you reorder 2nd eq.

[ocean seal]

PBj4FPxrZF7ovZ

[lapis harness]

k is your scaling factor

A' is scaled area

pretty intuitive I'd say

[alpine sable]

idk what you're trying to say, lol

[night sinew]

I need to find cos(x-y), i already have cosx. but i'm having trouble plugging in cosy into cos(x-y), as i don't know if the negative in cos(x-y) will cancel out the negative of cosy= -4/5

[shell widget]

@night sinew Just use the formula cos(x-y) = cos(x)cos(y) + sin(x)sin(y)

you have sin(x) = -1/4, but think of it as sin(x) = 1/4, make a right angled triangle, find the missing side and from there, you can find cos(x)

[night sinew]

yeah that's what i'm supposed to do and what i'm doing BUT

for cos(x-y), if i plug in cosy = -4/5

will the -y turn into a positive?

since -(-4/5)

[shell widget]

Don't think of it that way

you have sin(x) = -1/4, can you find cos(x)?

and you have cos(y) = -4/5, can you find sin(y)?

[night sinew]

already got em. cos(x) and sin(y)

[shell widget]

ok what are they?

btw put the appropriate signs according to the quadrants

[night sinew]

cosx is square root 15 / 4

sin(y) is 3/5

yes, since its in quadrant III-

sin and cos are negative.

[shell widget]

ok so cos(x) = -sqrt(15)/4, sin(y) = -3/5

thats what u got?

[night sinew]

yes

[shell widget]

ok now just plug them in cos(x-y) = cos(x)cos(y) + sin(x)sin(y)

we get cos(x-y) = -sqrt(15)/4 + (-1/4)(-3/5)

and simplify now

[night sinew]

quick question

since cosy = -4/5

will it remain a negative, even in quadrant III?

[shell widget]

yes

[tawny condor]

can I ask a question here?

or ur not done

[shell widget]

we are done

[tawny condor]

ok

[night sinew]

i wasnt done lol but go ahead

[shell widget]

oh really?

mb

[night sinew]

no its good, go ahead

[tawny condor]

thanks

AG = d, <HED = beta, <A = 2alpha, ABCD is a deltoid, CO = m, and <C = 90.
I need to find the area of triangle AFH in terms of alpha, beta and d.
(That's only part 1 of the question, and in part 2 I need to use m aswell so I think m isn't related to part 1)

[shell widget]

do ask if u have any more questions

[tawny condor]

btw my question is in trig

[shell widget]

@tawny condor i think try #geometry-and-trigonometry

[tawny condor]

but thats not for questions

[shell widget]

why not?

[tawny condor]

Idk

[shell widget]

it is

[tawny condor]

this is more of a questions place

this isn't restricted to any field

asking here is fine

[night sinew]

i think he meant you to try in that channel, so that you have a higher chance of getting the question answered since its more related to the topic

[night sinew]

wouldnt -4/5 , -1/4 and -3/5 all turn positive?

[shell widget]

yes

[tawny condor]

alr

[shell widget]

you get sqrt(15)/4 * (4/5) + (1/4)(3/5)

[night sinew]

alright, i got -sqrt(15)/20

[night sinew]

but can i ask about this?

is it STILL ok to do it like how I said it?

for cos(x-y), if i plug in cosy = -4/5
will the -y turn into a positive?
since -(-4/5)

[harsh swallow]

what is it called when it's not flat

positive and negative

curvature

sorry

i interrupted

[night sinew]

no its ok

[harsh swallow]

what's the problem, maybe i can help

[night sinew]

the whole problem is already answered but it's just one thing that's bothering me

[harsh swallow]

alright

[slow pendant]

guys how does one determine what type of conic it is

by the general equaton

[alpine sable]

i normally just try to convert it to a form that makes it clear

but you could also follow certain rules by looking at coefficients

you can find more here: https://en.wikipedia.org/wiki/Conic_section

[wispy elm]

Ok so I stumbled on a something and I wanted help understanding it.
I have a rule to generate a triangle of numbers. The first few rows look like this:

1
1, 1
1, 3, 1
1, 7, 6, 1
1, 15, 25, 10, 1

If k_n is the nth number of the kth row, the the rule is (k+1)_n = n*k_n + k_(n-1) and (k+1)_(k+1) = 1. In other words, you generate the next list by multiplying each element of the current list by its position in the current list, then adding the previous element in the list. Finally, you append an extra 1 to the end of the list.

It seems like adding up the numbers in the rows of this triangle gets you the Bell numbers (albeit missing B_0). I really don't know anything about this area of maths, so could anyone explain why partitions would come up here, or what property of the Bell numbers makes this happen?

[harsh swallow]

$(k+1)n = n*k_n + k{n-1}$

[ocean seal]

Katharine

[harsh swallow]

$(k+1)_{n+1} = 1$

[ocean seal]

Katharine

[vale wigeon]

i have a feeling that this notation is going to get very bad very quickly

[alpine sable]

Q and R dont have the same cardinality.
But how would I come to that conclusion?
I'd have to try every single function Q to R and R to Q in math and check that none is bijective?

[wispy elm]

I think the standard way would be to show that Q is countable by writing it in that big matrix, then show that R is uncountable using diagonalization. That should imply that they have different cardinality, right?

[ebon quest]

Quick question, how would I go about following this? I would really like some help on the steps involved

[manic quail]

At first, I would convert the units such that they fit.

Because "mi^2" means square miles, I presume, and km^2 square kilometres.

[ebon quest]

Yes, km^2 is square kilometers

[manic quail]

Once you have converted, you multiply density by area.

[carmine ermine]

Guys is dy/dx a fraction?

[manic quail]

Jaytron, don't start a war.

[carmine ermine]

No seriously

I'm being sincere

bc I keep getting it confused

[manic quail]

While technically you can treat it as one, it isn't a fraction.

[ebon quest]

So I convert them all into the same unit first, right?

[manic quail]

Yes.

[carmine ermine]

But how not, dy is supposed to equal the infinitely small difference in terms of the function

and dx is supposed to equal the infinitely small difference in terms of simple units or time [physics]

So to find the average rate of change

you divide the two

But since we can't do that

we put dy/dx
that's how I think of it

[manic quail]

Yes, and dy/dx can be seen as the very small change in y divided by an incredibly small change in x

[carmine ermine]

so how wouldn't that be a fraction..?

Not saying it's a third grade regular fraction

but it has to be some type of special case fraction

[manic quail]

because infinity.

[tranquil tulip]

what math is this

[carmine ermine]

Calculus

[ebon quest]

Okay so, so I convert and multiply the density by the area with both County A & B?

[manic quail]

Yes, @ebon quest .

[ebon quest]

And from there, that’s it? Correct? I’m just making sure bc it seems easier than I thought lol

[alpine sable]

Jaytron, your idea is an intuitive handwavy way to develop this stuff and intuition doesn’t really belong here

[manic quail]

@ebon quest , yes, that's all.

[ebon quest]

Cool! Thanks!

[carmine ermine]

Then where?

[alpine sable]

Intuitive ideas about infinity and infinitely small thing can create absurd results

[carmine ermine]

Well like always, I blame the ones who came up with the idea to place dy and dx as a fraction

[alpine sable]

Those people were quite helpful in the development of calculus actually

[carmine ermine]

They are but that idea confuses me, I'll put some more research into it and ty

[alpine sable]

Because it happens to be the case that if you treat them as fractions, you usually won’t run into problems

[carmine ermine]

Then lemme ask you this

[alpine sable]

But that requires proof

[carmine ermine]

What makes something a fraction?

[alpine sable]

Differential forms are what you are interested in if you want to research this

[carmine ermine]

Ohh alright

[carmine ermine]

Ty

[alpine sable]

Well you’ll probably have to specify “a fraction of real numbers”

But it’s probably clear what is

An “infinitely small change in a variable” is not a real number so dy/dx is not a fraction of real numbers that we can automatically assume follows all the usual algebra rules for the real numbers

[swift widget]

Guys whats the antiderivative of 1/2 ??

wolframalpha says its 1/x ???

[manic quail]

1/2 x + c, @swift widget

,w integral 1/2

[ocean seal]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by veryhappyperson#1152

[swift widget]

https://www.wolframalpha.com/input/?i=integral+(x^2%2F16%2B1%2Fx^2-1%2F2)

Is that the correct antiderivative of that expression=

?

[manic quail]

yes.

they just swapped the terms around.

[swift widget]

Okey. Yeah, im tired. thanks for confirmation

haha

[random latch]

can anyone help me with this one, i cant think >.<

[ionic jewel]

if two typists type two pages in two minutes, that means a typist types one page a minute

[random latch]

this is what i got, but my friend got 6, who is right

my friend:

https://cdn.discordapp.com/attachments/805876887711645727/833032918641278976/20210417_183437.jpg

[ionic jewel]

wait what's your actual answer tho?

[random latch]

mine is 3, friend is 6

[ionic jewel]

yours is right

[random latch]

my logic is that
2 typists = 2 pages / min
so 1 typist = 1 page / min
so 1 typist + 1min = 1 page
so 6 min + 1 typist = 6 pages
so 3 typists

[ionic jewel]

yes that's exactly right

[random latch]

why is my friend wrong

18 pages = 36 typists / min
and so 6 typists / 6 min right

[ionic jewel]

looks like s/he somehow got 1 typist types half a page in a minute

6 typists for 6 minutes would make 36 pages yes

wait a second

i think i read it wrong too

lmao

your friend is right lol

you simplify the first part to "one typist can type one page in two minutes"

@random latch

[random latch]

omfg, i read it as 2t = 2p / min not per 2 min

thank you so much!

[ionic jewel]

haha yeah i got confused too, nw

[versed osprey]

Hey guys, I don’t really understand how I’m supposed to do 22 and 23, since they seem to have the same logic

[versed osprey]

For 22, I’m able to solve the 1st part and I’m not able to solve the second part. For 23 I can’t solve both parts

[ionic jewel]

how did you do 22 part 1?

[glass lichen]

$x^2-\frac{1}{x^2}$ is difference of squares if you did i

[ocean seal]

moshill1

[unreal forum]

what is 420x69

[versed osprey]

Yo nvm nvm, I found part 2 by finding part 1

It’s just the 23 one

[ionic jewel]

looks like the opposite of 22

[unreal forum]

what is 420x69

[versed osprey]

But I put the expression to the power of two, subtracted 2 from both sides. And got the part two answer using that.

[unreal forum]

Fuck You.

[worldly island]

https://i.imgur.com/icuWXOf.png how do i find the second t value? you use the y=5sin2t and make it equal to 4.2
so therefore 2t=arcsin4.2/5 but how do you get the 2nd t value

nvm worked it out

[limber atlas]

can someone expand how this expands

[sly mantle]

read it as (a+b)c=ac+bc

[limber atlas]

ohhhhh

thank you!

[spice crypt]

Quick question: Does the trace of a linear transformation (from R^n to R^n) represent anything specific to the transformation itself?

The determinant represents the proportionality of the areas (I think) from the input space to the output space. I'm looking to see if the trace characterizes something about the transform itself rather than just being the sum of the eigenvalues

[alpine sable]

it doesn't have a geometric interpretation as common and important as a determinant

but there are some ways to interpret it

maybe look at https://mathoverflow.net/questions/13526/geometric-interpretation-of-trace

[spice crypt]

Cool, thanks!

[grave belfry]

I have been stuck at this problem for a while

I just dont understand how sum of infinite gp and be the common ratio itself

[sage summit]

I think the question wants to say this : let a be a real number and suppose there exists two different real numbers r1,r2 such that the infinite series a + ar1 + ar1² + ... sums to r1 and the infinite series a+ar2+ar2²+... sums to r2. Then determine r1+r2 in terms of a

[grave belfry]

Its numerical
The ans is 1 somehow

[sage summit]

ah yeah

[glass lichen]

$+\infty$

[ocean seal]

moshill1

[glass lichen]

[sage summit]

yeah the question is very

[grave belfry]

It's the Q's fault right...

I mean, is it wrong

[sage summit]

to write "+ infinity" like that ? yeah idk what was in the writer's head when they did that

[grave belfry]

XD

[glass lichen]

$\frac{a}{1-r_1}=r_1 \ \frac{a}{1-r_2}=r_2$

assuming i have the formula right, might be a minus?

[grave belfry]

Its minus

Ik that, how how do u deal with 'a'

[ocean seal]

moshill1

[glass lichen]

$r_1+r_2=a\left(\frac{1}{1-r_1}+\frac{1}{1-r_2}\right)$

[ocean seal]

moshill1

[grave belfry]

?

I still dont get how do u cancel a

[indigo jetty]

$a = (1-r_1)r_1 \ , a = (1-r_2)r_2$

[polar quest]

O ne amk

[ocean seal]

Ken G

[grave belfry]

Ooo
Got it
Tnx everyone

[alpine sable]

hiiii

i feel dumb rn but is anybody good with probabilities? i have some homework questions and idk how to do em

the two i need help with are
“Find the probability of rolling two dice and getting a sum of at least 10.”

“Find the probability of flipping a coin 9 times and getting at most 3 heads, rounded to the nearest percent.”

There are 36 ways to rolls two dice, each way with equal probability

How many of them result in a sum of at least 10?

[limpid spade]

A={(5,5), (6,4),(4,6), (6,5),(5,6),(6,6)}

Dunno if 1 (5,5) or 2

Either way it's 4/36 or 3/36

[alpine sable]

Just 1, and you’re missing a case

[limpid spade]

At least 10 so it can be bigger than 10

[alpine sable]

How about (6,6)

[agile compass]

It's easiest to make an addition table

[limpid spade]

6/36

1/6

@alpine sable u understand how I came to that?

[alpine sable]

Ummm

thank you guys i got em down!!! also @alpine sable i love your pfp

I was asking Ezra the question

also i have one more question but it’s discrete math

Not asking for help with it

And lol thanks

[limpid spade]

Do u understand how I got the answer?

[alpine sable]

i think so yeah, i found a similar answer online using different numbers and that broke it down too

[limpid spade]

Okie

[alpine sable]

this ones weird i should’ve paid attention in class i don’t think it’s something ppl typically learn idk

#36

also @alpine sable i love ur pfp

You can look up NNA if you have not seen it and easily determine that the answer is b

And thanks (again) lol

oh wait did i already say it 💀💀 this math is literally frying my brain

Yes

LMAO

oops

i’ve been doing homework all day i think i should take a break

alright well that’s all i needed thanks guys :))

[somber spoke]

<@&286206848099549185>

i keep getting 17b as 1

but apparently it's 23 according to the markscheme

[night blade]

wait 15 minutes, read the rules

[somber spoke]

oh sorry

[austere slate]

can someone help me with integrals

easy level i guess but im getting confused

[alpine sable]

whats the problem

[somber spoke]

nvm i misread the question

[bold tusk]

if a series is absolutely convergent, can it have all negative terms?"

or are all the terms positive?

[alpine sable]

it can

consider the sequence $a_n = - \frac{1}{2^n}$

[ocean seal]

Matejp1

[bold tusk]

ok so another question

when doing the limit comparison test

can both a_n and b_n be negative?

or does it work when both of them are positive?

[austere slate]

this...idk what im doing wrong

i must use substitution to solve this integral

[alpine sable]

i think you just have to plug t=1-x in the end

[austere slate]

the what

💀

[bold tusk]

and dont forget the +C coz its indefinite

[austere slate]

yes i know

[bold tusk]

you said t = 1 - x

[austere slate]

see we solve it like this

[bold tusk]

after solving it, you have to sub it back in

[austere slate]

its not working this time idk why

[austere slate]

i know

but my problem is in the answer i shouldnt get the -

[alpine sable]

when integrating the answers can differ by a constant

your answer is correct

[bold tusk]

you can check by opening up the brackets and integrating each term and youll still get a negative answer

[alpine sable]

try to differentiate your answer and the formal solution

and see which one is correct

[austere slate]

wait

[alpine sable]

normally the test is stated as requiring all terms of a_n to be non-negative and all terms of b_n to be positive

but...

if they are both negative, you can essentially still use the test

[austere slate]

oh shit

its correct

[bold tusk]

thank you!

[austere slate]

how am i so dumb

@alpine sable @bold tusk thanks

[bold tusk]

dw it happens to the best of us

[alpine sable]

the only trouble will be if either of the series have some negative and some positive terms

[bold tusk]

i know that a_n is absolutely convergent and that b_n is a multiple of a_n

and they are both the same sign

[alpine sable]

a series $\sum_{n=1}^\infty a_n$ converges if and only if -$\sum_{n=1}^\infty a_n$ converges

[ocean seal]

Botnuke

[alpine sable]

hey what is actually limit comparison text

english isnt my native language

[bold tusk]

if $a_n \geq 0$ and $b_n > 0$, and $\lim_{n \to \infty} \frac{a_n}{b_n} = L < \infty \neq 0$, then both of the series either converge or diverge

[alpine sable]

oh nvm i searched it on google

but its surprising we havent covered it

[austere slate]

imma throw my notebook out of the window

im getting an extra -1/3

[ocean seal]

TheGoner

[alpine sable]

we only did root, ratio, integral and raabes test

does this limit comparison test allow us to show convergence in cases when the other 4 tests fail?

or is it weaker?

[alpine sable]

if you're getting an extra constant thats absolutely fine. That's why there is +C, because they can differ by a constant.

it can work if those won't be helpful

[austere slate]

oh-

[bold tusk]

LCT is usually done on polynomial limits

[alpine sable]

oh thanks i'll look into it a bit more

[alpine sable]

limit comparison test might not be a good idea

what's the problem?

wait, couldn't you then just factor the number it's multiplied by out?

and if a_n is convergent, so is b_n

∑a_n being absolutely convergent does not mean a_n ≥ 0 for all n

[bold tusk]

$a_n \neq 0, \forall n \geq 1$, and $b_n = c * a_n$, where $c > 0$, prove that $b_n$ is absolutely convergent

[alpine sable]

i'm not an expert but i'd try to prove that by definition

[bold tusk]

the sum starts from n = 1 to infinity

[ocean seal]

TheGoner

[alpine sable]

do you also know ∑a_n is absolutely convergent?

[bold tusk]

yes

[alpine sable]

I also agree with proving this from the definition

it follows readily from limit and sum properties

ugh my latex skills

yes limit of partial sums of a_n exists iff limit of partial sums of b_n exists

[bold tusk]

ohhh thank you

i completely forgot about the definitions

[manic quail]

@oak chasm , if I may ask again, is there a fancy Mathematica way to get the Fibonacci numbers smaller than 4 000 000?

Or is looping and checking if they are less than 4 000 000 the only way?

[alpine sable]

Can someone please help me with a logic exercise?

[oak chasm]

@manic quail Something like ```
TakeWhile[Map[Fibonacci, Range[1, 4000000 + 1]], # < 4000000 &]

[manic quail]

I just did it the classic way for now, thanks though.

[oak chasm]

No problem.

[alpine sable]

pretty confused on where to start

[timber belfry]

$$
1800 \cdot 1.02^n = 1800 \cdot 2\newline
\Leftrightarrow 1.02^n = 2\newline
= e^{\ln (1.02^n)} = e^{n\cdot\ln (1.02)} = e^{\ln (2)}\
\Leftrightarrow n\cdot \ln (1.02) = \ln(2)\newline
\Leftrightarrow n = \frac{\ln(2)}{\ln(1.02)} = 35.00278879...
$$

[ocean seal]

𝔸𝕟𝕕𝕣𝕖𝕒𝕤

[gray gorge]

$a\b$

[ocean seal]

(R / I) / (J / I)ttgenstein

[gray gorge]

double backslash

[ocean seal]

𝔸𝕟𝕕𝕣𝕖𝕒𝕤

[gray gorge]

Did you use a double backslash? \\

Are you trying to write $\begin{bmatrix}a\b\end{bmatrix}$?

[ocean seal]

(R / I) / (J / I)ttgenstein

[alpine sable]

I'm on this one now sorry

[gray gorge]

$$
1800 \cdot 1.02^n = 1800 \cdot 2\
\Leftrightarrow 1.02^n = 2\
= e^{\ln (1.02^n)} = e^{n\cdot\ln (1.02)} = e^{\ln (2)}
\Leftrightarrow n\cdot \ln (1.02) = \ln(2)\
\Leftrightarrow n = \frac{\ln(2)}{\ln(1.02)} = 35.00278879...
$$

[ocean seal]

(R / I) / (J / I)ttgenstein

[gray gorge]

Discord renders only one backslash because backslashes are used in the chat window as command syntax themselves

or was it forward slashes

anyway yeah it's an issue with Discord

$1+1\=2\=3-1$

[ocean seal]

(R / I) / (J / I)ttgenstein

[timber belfry]

Dammit

[gray gorge]

$
1800 \cdot 1.02^n = 1800 \cdot 2\
\Leftrightarrow 1.02^n = 2\
= e^{\ln (1.02^n)} = e^{n\cdot\ln (1.02)} = e^{\ln (2)}\
\Leftrightarrow n\cdot \ln (1.02) = \ln(2)\
\Leftrightarrow n = \frac{\ln(2)}{\ln(1.02)} = 35.00278879...
$

That doesn't even render

lmao

Well if you wanna mess around with the LaTeX typesetter, feel free to do that in #bots

[ocean seal]

𝔸𝕟𝕕𝕣𝕖𝕒𝕤
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[timber belfry]

Oh, you right

I will delete what I have written.

But thanks

[bronze trench]

i'm so confused

no idea

[jade birch]

Thales theorem?

[bronze trench]

No I don't think so

isn't that with a circle

[jade birch]

Why not?

[jade birch]

And also with a triangle

Your radius is 4

Form a new triangle

Like this

Let me draw it

[bronze trench]

ok

[jade birch]

With hypothenuse length 4 and one side length 2

Find the height

And then find x with pythagoren theorem

[bronze trench]

ok I see, thank you!

[jade birch]

No problem

[dry echo]

ok so my college uses http://mymathlab.com for homework

This is a screenshot from there

I also use symbolab.com to check my results sometimes

This is the symbolab result at the bottom

Is www.symbolab.com making mistakes or the homework website?

[knotty sleet]

You missed a factor of 2

[dry echo]

OHHH waitt

[knotty sleet]

🙂

[dry echo]

I am dumb

[knotty sleet]

XDD

[dry echo]

Thanks

[knotty sleet]

Np we all have those moments

[proper perch]

Find the 16 codewords in the Hamming code of order 3 described in Example 18.

[carmine ermine]

Can someone tell me what this is?

Like what is that in calculus when two constants are on the top and bottom side of the right side of a variable?

[alpine sable]

one is an exponent one is a subscript?

y_1 is just a variable name

just like y is a variable name

or x

or x_5

[carmine ermine]

Oh

So what does that example mean?

[carmine ermine]

.

[alpine sable]

not sure without the context

looks like $y_1^4$?

[ocean seal]

Botnuke

[carmine ermine]

Yeah

[alpine sable]

oh, so it's not an exponent here, actually

y_1 is the variable name though

y'(x) is the first derivative of y(x)

or, $y'(x) = \frac{dy}{dx}$

[carmine ermine]

Ah

[ocean seal]

Botnuke

[alpine sable]

by definition

it's just notation

[carmine ermine]

So we're treating y like a function?

so

why is there a one below the y

[alpine sable]

and $y''(x)$ is the second derivative

[ocean seal]

Botnuke

[alpine sable]

well that's just a naming convention they are using

they named the function $y_1$ rather than $y$

[ocean seal]

Botnuke

[alpine sable]

maybe they will introduce another function named $y_2$ later

[ocean seal]

Botnuke

[alpine sable]

and maybe y_3 and y_4

[carmine ermine]

OHHHHHH

OMG

Tysm!!!!!

I actually understand what they're doing

[median yoke]

Is this the calculus help group?

[alpine sable]

these question channels don't have any particular guidelines

math content guidelines I mean

[sly mantle]

@silent vapor do you think this is true?

[bold tusk]

how can you check if a function is equal to its maclaurin series?

[sly crane]

Hello, I have a problem I need to solve for a software Im working on that only allows sum, substraction, multiplication and division

How can I get the absolute value ?

actual problem is, I need the sign of X, meaning I need +1 or -1 depending on the sign of X

I can do this if I had abs, I would do: x/abs(x)

but I dont have that operation

[bold tusk]

just use an if statement that returns -1 or 1 depending if x is less than 0 or not

[sly crane]

no, I dont have conditionals

it only allows sum, substraction, multiplication and division

thats what the software can parse

[bold tusk]

$\frac{x}{\frac{x^2}{x}}$

[ocean seal]

TheGoner

[sly crane]

hmmm, let me process that

[bold tusk]

wait

nvm

that doesnt work

are you able to take powers?

or just those 4 operations?

[sly crane]

I can manuallyy ad them

using *

[dire wren]

you can type xx for x^2

[sly crane]

yeah

[sly crane]

yeah doesnt work, that will always be +1

[neon olive]

im stumped on this problem

The end of each blade of a ceiling fan is two feet from the center of the fan and makes three full rotations per second. Given that there are 5280 feet in a mile, what is the speed, in miles per hour, of the end of one of the blades? Express your answer as a decimal to the nearest tenth.

[wispy elm]

Is the problem calculating the velocity, or the unit conversion?

[neon olive]

idrk cause im doing this cause my sister doesnt know im in 9th grade and my sister is in 6th but she has this elective where it gives her homework on algebra 2 and stuff

i think its calculatin the velocity

calculating*

[dapper umbra]

they asked to calculate speed.

use the formula for it

[neon olive]

idek the formula i havent even learned this yet im learning algebra this algebra 2

[dapper umbra]

speed = distance/time

[neon olive]

oki

[dapper umbra]

althought since it says rotations, I think u have to use a diff formula

[neon olive]

???

anyone here???

ig not

@dapper umbra is the topic of this problem linear speed and angular velocity

anyone

[wispy elm]

Speed is distance/time

If the fan makes 3 rotations each second, the distance it covers in that time is 3 times the circumference of the circular path it's following.

We are given that this circular path has radius 2 feet, so the circumference of the path is 2*pi*r = 4pi feet. 3 times that gives us 12 feet per second. The rest of the question is a matter of converting to miles per hour.

[neon olive]

ok

@wispy elm sorry for wasting ur time i wouldnt have helped my sister but she made me a offer where she gave me all her money

so i have learned this yet cause im in 9th grade so im learning algebra

[wispy elm]

Everything you do on discord is a waste of time in a way, so it's no issue.

[neon olive]

true i spend approx 10 hours on dc so

https://tenor.com/view/memes-risitas-laughing-lmao-lol-gif-11500030

[thorn bobcat]

,iam talks

[ocean seal]

Gave you the Talks selfrole.

[gaunt magnet]

https://tenor.com/view/cat-leaving-leave-oops-bye-gif-17519616

[tribal geyser]

Can someone pls help with 20

I’m not getting 4

I let the roots be alpha alpha and beta

Pls help

[alpine sable]

think I got it

we have $\alpha^2\beta = 18$, and a substitution into $\alpha^2+2\alpha\beta = -3$ leads to $\alpha^3+3\alpha+36=0$. Then $\alpha = -3$ by some machinery which may not be allowed. Back to $\alpha^2\beta = 18$, we then have $\beta = 2$. And finally $2\alpha + \beta = -m$ implies $m=4$

[ocean seal]

Botnuke

[tribal geyser]

Um I got what u got but idk how to work out alpha lol

[alpine sable]

this is probably not what the book intended

but hmmm I don't see a way to get around it

[paper edge]

My first thought process is the rational root theorem where a_0 = -18 and a_n = 1. Thus, the rational roots must be in +/- {1, 2, 3, 6, 9, 18}

[alpine sable]

yea actually that is probably what you are expected to do

12 divisions though

[paper edge]

Yeah...pain

[alpine sable]

I guess as soon as you got -3 you could role with it and assume there is only one m based on how the problem is written

but to show m = 4 is the only solution you'd have to do all 12 divisions lol

actually you don't need to do divisions

[paper edge]

Somewhat true. You don't have to test all 12 divisions, but you do have to test all 12 numbers to see how the polynomial would equal 0.

[alpine sable]

yea

[paper edge]

Big brain mode

[tribal geyser]

Yh I don’t get it lol

[alpine sable]

have you heard of rational root theorem before?

[tribal geyser]

Yes but how is that suppose to help in this situation

[alpine sable]

actually I don't think that will work because we don't know m yet

we have to do rational root theorem on $\alpha^3+3\alpha+36=0$

[ocean seal]

Botnuke

[tribal geyser]

Oh yh to do long division?

[alpine sable]

no scratch that idea

[tribal geyser]

Oh

so

[alpine sable]

the only possibilities for α are divisors of 36

[tribal geyser]

-3

[alpine sable]

yes

[tribal geyser]

So m is 4

[alpine sable]

you could go through all the rest of them if you wanted to show m = 4 is the only possibility

[tribal geyser]

but how do we know there is no other values

Oh

is this the only wayC

Way*

[alpine sable]

no, computer algebra can give you all 3 roots of $\alpha^3+3\alpha+36=0$, and $\alpha = -3$ is the only rational one

[ocean seal]

Botnuke

[tribal geyser]

Ohhh

Thank you so much!!

[alpine sable]

but to do it by hand, yes, exhaustion by rational root theorem is probably your best bet

cubics are not very nice things

[tribal geyser]

yepp thank you😊

[tribal geyser]

Ohh

[alpine sable]

that's how I would do it lol

[tribal geyser]

Sorry could I ask u another similar question

[alpine sable]

sure

[tribal geyser]

I approached it like this

So if two roots is 0 then I let alpha be the other root

meaning alpha + 0 = 1/3?

[alpine sable]

yes

[tribal geyser]

then how to do I work out the other roots?

[alpine sable]

factor out $x-\frac{1}{3}$

[ocean seal]

Botnuke

[alpine sable]

then you'll have a quadratic which is easier

[tribal geyser]

Umm could I not do it another way?

Using sum and products

[alpine sable]

hmm yea you probably can

[tribal geyser]

How would I do that lolol

[alpine sable]

name the roots $\alpha$, $\beta$, and $\gamma$ and suppose $\alpha = -\beta$

[ocean seal]

Botnuke

[alpine sable]

we know $\gamma = \frac{1}{3}$

[ocean seal]

Botnuke

[tribal geyser]

wait how is alpha equal to -beta

[alpine sable]

2 of the roots sum to 0

I'm letting α and β be those roots

[tribal geyser]

Ahh I see

Ok

Yep and then

[alpine sable]

we also have... $-\beta^2 - \frac{1}{3}\beta + \frac{1}{3}\beta = -\frac{1}{3}$?

[ocean seal]

Botnuke

[tribal geyser]

Um

[alpine sable]

from $\alpha\beta+\alpha\gamma+\beta\gamma=\frac{c}{a}$

[tribal geyser]

How did u get that sorry

[ocean seal]

Botnuke

[alpine sable]

check it though, I'm not being very careful

[tribal geyser]

How did u get beta squared

[alpine sable]

α = -β

[tribal geyser]

oh coz I thought alpha beta=0

whoops

[alpine sable]

so if that's right we have $\beta = \frac{1}{\sqrt{3}}$, which implies $\alpha = -\frac{1}{\sqrt{3}}$

[ocean seal]

Botnuke

[alpine sable]

and then we are done

[tribal geyser]

So ur saying the other root is?

[alpine sable]

these are the other 2 roots if my other relation is correct

[tribal geyser]

The roots in the answer is 1/3, plus or minus 1/2

[alpine sable]

alright I made an error then

most likely

[tribal geyser]

This is confusing lol

[alpine sable]

oh haha it's because 3/12 is 1/4, not 1/3

so what we really have is $-\beta^2 - \frac{1}{3}\beta + \frac{1}{3}\beta = -\frac{1}{4}$

[ocean seal]

Botnuke

[alpine sable]

which implies $\beta = \sqrt{\frac{1}{4}}=\frac{1}{2}$

[ocean seal]

Botnuke

[alpine sable]

and so $\alpha = -\frac{1}{2}$

[ocean seal]

Botnuke

[alpine sable]

do I need to explain it again?

[tribal geyser]

Ohhh

Okay thanks!!

@alpine sable

[alpine sable]

no problem

[amber ermine]

If the channel is open, I would like help with the following:

Need help with the following problem: Find the laplace transform of the function f(t) = (2-t)(h(t-2)-h(t-6)) ? I have tried simplifying the second quantity, which then gives me (2-t)(4h), then I distributed that, and used e^(-st) with each piece when doing the integration portion. I get something like -8h-4ht as my final answer, however this is not correct. Where am I going wrong? Also, "s" is assumed as non-zero

[amber ermine]

<@&286206848099549185> Please see the preceding and help if able. If my question is not clear please let me know

[feral swan]

if 5 of X is £2
how much would 2000 of X be?

[ionic jewel]

5x=2

2000x=?

[feral swan]

i figured it out its 540 nvm

thanks @ionic jewel

[ionic jewel]

that doesnt sound right

[feral swan]

what

. _.

[ionic jewel]

,calc 2000(2/5)

[ocean seal]

Result:

800

[feral swan]

oh

damn

[silent nest]

If i have the volume x and want to distribute it to a volume of ratio 1:1:6

how do i calculate the w x l x h in terms of x

[willow lake]

prove for any 3d convex polyhedron that any vertex has a minimum of 3 edges

[alpine sable]

nitro + nitro?

https://discord.gift/S7tsCPktaFG7ZWwJ

helppppp

im unsure how to answer this question, could anyone help?

[vale wigeon]

@alpine sable is there a particular one among these 18 subproblems which confuses you?

or are you just confused n general

[ocean seal]

DeadPhysicist

[vale wigeon]

bad tex

[idle plover]

Beginner

[vale wigeon]

$\sin(93^\circ) = \sin(90^\circ + 3^\circ) = \cos(3^\circ)$

[idle plover]

How do we put the degrees?

[ocean seal]

Ann

[idle plover]

Ou

[vale wigeon]

clinx doesn't seem all that eager to respond though

or maybe they're just away

[idle plover]

Perhaps they're away

[vale wigeon]

posting a question in #help-0 and then disappearing is a surefire way to get it buried

[alpine sable]

@vale wigeon

im sorry im on dnd

[vale wigeon]

dnd still makes ping icons show up does it not

[alpine sable]

I dont understand how to answer a

[vale wigeon]

in all of these problems you'll need to make use of symmetries of the trig functions

if you want i can write out all the relevant ones in a list

[alpine sable]

ok

thanks

[vale wigeon]

here you go @alpine sable

[alpine sable]

How do I solve for theta when $\tan ^2 \theta = \frac{1}{3}$

[ocean seal]

Jeed_for

[winter bay]

Take the square root of both sides and then the inverse tangent

[alpine sable]

I only know how to do it if it were an angle I knew like sin^2(2theta) = 1

ohhhhhhh

I completely forgot about inverse functions

[winter bay]

So $\tan{\theta} = \frac{\sqrt{3}}{3}$

[ocean seal]

visual of Petter's ascendance

[alpine sable]

wait but these questions are technology free

[winter bay]

?

Idk what u mean

[dusty sun]

hi i have problem in rational numbers
I am not able to do rational numbers

[alpine sable]

Well you cant do an inverse tangent by hand can you? We aren't allowed to use calculators for these questions

I dont really know how the inverse functions work, the only time we ever did it was with basic trig a couple years ago

[idle plover]

$tan(\theta) = \frac{1}{\sqrt{3}}$

[winter bay]

Well

[idle plover]

Oops

√3

Typo

[winter bay]

Inverse functions are kind weird

[vale wigeon]

±sqrt(3)/3.

[ocean seal]

DeadPhysicist

[winter bay]

U could solve it more mathematically by remembering that tan theta = sin theta / cos theta

[vale wigeon]

unless you want to miss a whole swath of solutions you will need to solve both $\tan(\theta) = 1/\sqrt{3}$ and $\tan(\theta) = -1/\sqrt{3}$

[ocean seal]

Ann

[winter bay]

Oh

Yeah thats true

I always forget plus minus when taking the sqrt

[vale wigeon]

also, "more mathematically"? there is absolutely nothing "not mathematical" about going from tan^2(θ) = 1/3 to tan(θ) = ±1/sqrt(3).

[winter bay]

Yeah idk why I said that

[alpine sable]

Cause im not sure if we are meant to use inverse fuctions, the other questions are pretty simple, like $\sin ^2 2 \theta = 1$ where I just knew $2x^2$ had to equal 90 and solved it algebraically

[idle plover]

For theta being an acute angle, we can take the positive value if I'm not mistaken

[winter bay]

What I meant was avoiding using a calculator to find inverse tangent of plus minus 1/sqrt 3

[ocean seal]

Jeed_for

[vale wigeon]

2x^2?

[alpine sable]

@vale wigeon ty!

well 2 theta^2 = pi/4*

[vale wigeon]

why are you squaring theta?

[alpine sable]

im saying that was how I worked out previous questions which didn't require inverse functions

the question for that example was $\sin ^2 2 \theta = 1$

[ocean seal]

Jeed_for

[vale wigeon]

you're claiming that $\sin^2(2\theta) = 1$ can somehow be transformed into $2\theta^2 = \frac{\pi}{4}$, which as-is sounds like complete bogus to me

[ocean seal]

Ann

[alpine sable]

pi/2*

[winter bay]

I'm interested to see how it could...

[vale wigeon]

even replacing the pi/4 with pi/2 won't change the bogus-ness of this transformation

why on earth is THETA getting SQUARED all of a sudden???

[winter bay]

Bc squares are fun

[alpine sable]

I think you've misunderstood what I was saying, sin^2(2theta) = 1, 2theta^2 must equal pi/2

as sine(pi/2) is 1

[vale wigeon]

why 2θ**^2**????

why are you squaring theta!!!

i can see it loud and clear, you're saying $2\theta^2 = \frac{\pi}{2}$!

[alpine sable]

because the original question squares the answer to sine(2theta)

[ocean seal]

Ann

[vale wigeon]

this would make $\theta = \pm\frac{\sqrt{\pi}}{2}$

[ocean seal]

Ann

[alpine sable]

which is the answer I got

[winter bay]

I think the confusion is that the whole function is being squared not specifically theta

[alpine sable]

im solving for theta

[vale wigeon]

this answer is wrong

if you don't believe me:

,calc sin(2 * sqrt(pi)/2)^2

[ocean seal]

Result:

0.95988249738426

[vale wigeon]

not 1.

[alpine sable]

how would we do it then

[vale wigeon]

well you could turn $\sin^2(2\theta) = 1$ into $\sin(2\theta) = \pm 1$

[ocean seal]

Ann

[alpine sable]

? how

[vale wigeon]

the same way you would turn $z^2 = 49$ into $z = \pm 7$?

[ocean seal]

Ann

[alpine sable]

ohh ofc, right

[vale wigeon]

by taking the square root of both sides

[alpine sable]

so theta is plus or minus 45?

or pi/4

[vale wigeon]

±π/4 are just two out of an infinite family of solutions

[alpine sable]

yeah it says $\theta \epsilon [0, 2\pi]$ so I guess thats right

[ocean seal]

Jeed_for

[vale wigeon]

$\theta \in [0,2\pi]$

[ocean seal]

Ann

[vale wigeon]

there are other solutions in [0, 2pi] though

and -pi/4 isn't even there

if you want to restrict yourself to [0,2pi] there are four solutions:

pi/4, 3pi/4, 5pi/4, 7pi/4

[idle plover]

May I know your current qualifications? @vale wigeon

[vale wigeon]

...

i mean

ok

im a 4th year applied math student who does tutoring on the side

[idle plover]

It's fine if you don't wish to share, just curious

[vale wigeon]

is that what you wanted

[idle plover]

Oh yes I'm pretty sure that's what I wished to know

Thanks

Posting here as someone kind of decided not to abide by the rules and post right next to my question -

[ocean seal]

DeadPhysicist

[vale wigeon]

i have a feeling you've overcomplicated this somewhat

[upper mesa]

I'm so confused

what's going on?

[idle plover]

Oh maybe, any other approach?

[upper mesa]

especially this part

[idle plover]

Well, I posted my question, i.e. this question, in #help-1 but someone decided to post another question before anyone responded so I posted it here, hope that's fine

[upper mesa]

it's every man for himself

price of getting help for free I'm afraid

[idle plover]

Perhaps

[vale wigeon]

hm

weird

okay so it looks like my feeling wasnt quite warranted

[idle plover]

Oh so did you figure out anything?

[vale wigeon]

im trying to mess around with it but as of now i haven't arrived at anything different from you

[idle plover]

Oh alright, I thought of trying more numbers through trial and error but that'd be more or less of a waste of effort I think

[vale wigeon]

it's weird

i really don't feel like wading through a forest of algebra

it feels as if a really clever solution is just around the corner but i can't grasp it

[idle plover]

True, it's kind of tricky, I think there might've been a different approach but my approach does not seem wrong either

Being stuck at this for like a month by now

[vale wigeon]

a month? jeez

[idle plover]

Asked 4 teachers in total including a professor but well, no luck

[willow lake]

oh

[idle plover]

I'll be offline, if you don't mind, dm me if you figure out anything anytime, even if it's like a week or more later @vale wigeon

[vale wigeon]

ok one of you will have to move

you two posted your questions simultaneously

[empty forge]

36 * 36 * 36 *36 * 36 *36 *36 *10 *8
my thinking is that there 7 slots in which there are 36 possibilities (26 letters + 10 digits), the remaining slot has to be a digit (10), but we also have to multiply by 8 because the digit slot can be anywhere of the 8 characters.

I could be wrong, but its my honest attempt.

my classmates lives off redbull and cigarettes. Schools messes people up.

best of luck.

[alpine sable]

You might have missed the condition where the first character needs to be a letter

[empty forge]

I did forget about that

26 * 36 * 36 *36 * 36 *36 *36 *10 *7
Does this sound right?

[alpine sable]

Not sure, but for an answer I got 26*36^7 - 26^8

Which is: the number of passwords satisfying the first 3 conditions minus the number of passwords which are all letters

[woeful pulsar]

looks reasonable

[gray gorge]

[narrow wing]

can someone teach me how to solve euqiaotns using matrices

[gray gorge]

Can you give an example

[gusty light]

take a photo?

[narrow wing]

it just says use the matrix algorithm to solve:
2x + 1y = 5
3x + 2y = 8

[alpine sable]

so the area of MBC is 8cm2 and I need to find the rest, how do I calculate it?

i thought that considering the fact that M is the midpoint, that means that AD and BC are the same lenght as AM

So if i Multiply AM by 2 I could find the AB therefore letting me calculate the entire area

but i dont know the lenght of AD neither BC

[jade birch]

Can you show the whole question?

[alpine sable]

well its in italian

Look at the ABCD parallelogram. M is the midpoint of side AB. The area of ​​the MBC triangle measures 8 cm2.
How large is the area of ​​the parallelogram?

this is translated