help-0

[chrome salmon]

Yeah like that

[shadow stream]

ok let me try

Like this?

[noble sinew]

Not how method of transformation works

Start with finding range

[shadow stream]

what do you mean?

[noble sinew]

Support of Y

[shadow stream]

is it root(2) < y < root(3)?

i just plugged in 2 and 3

for the ranges

[noble sinew]

Si

then we have the following:

[shadow stream]

so i just integrate this between root 2 and root 3?

[noble sinew]

no we aren't doing any integrating

not how method of transformation works

[shadow stream]

oh

[noble sinew]

so we check the conditions are met for the theorem I posted

is it strictly monotone and differentiable?

[shadow stream]

it doesnt state 😦

[noble sinew]

you have a function

g(x)=sqrt(x)

so you tell me

[chrome salmon]

You have to determine

[shadow stream]

oh

im not familar with the jargon let me look up what monotone mean XD

[shadow stream]

ik its differentiable

not sure about monotone

[noble sinew]

,w plot sqrt(x) from 2 to 3

[ocean seal]

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[noble sinew]

seems monotone to me

[shadow stream]

does monotone mean you can inverse it?

im not really sure what monotone means

[noble sinew]

strictly monotone means strictly increasing or strictly decreasing

[shadow stream]

ohh ok

that makes more sense

so since both conditions are satisfied I would divide the original equation by the derivative of g(x)?

[chrome salmon]

https://math.stackexchange.com/questions/1746077/difference-between-increasing-and-monotone-increasing-function

[noble sinew]

yes (notice the x_1 though)

[shadow stream]

im not sure how to interpret the x_1

does it imply 2?

[noble sinew]

where g(x_1)=y

here is an example of the theorem used:

[shadow stream]

y = root(x)
then g(x_1) = root(x)
then x_1 = x?

[noble sinew]

no

[shadow stream]

so i inverse g(x)?

and x_1 = x^2

[noble sinew]

si

[noble sinew]

no

[shadow stream]

y = root(x)
inverse is not x^2?

[noble sinew]

you want to find a pdf of Y agree?

[shadow stream]

yes

[noble sinew]

why would we end up with a variable called x

in our pdf for Y

[shadow stream]

instead of x we would use y^2 then?

[noble sinew]

yes

[shadow stream]

ok

let me try to work it through

Like this?

[chrome salmon]

Ok

[shadow stream]

6/19 (y^2)^(3/2)

[chrome salmon]

2(3/2)=3

[noble sinew]

no you messed up somewhere

6/19 y^5 should you end up (can check yourself what you found wasn't a valid PDF (is integral from -inf to inf of f_Y(y) equal to 1?))

and anyway even if it was correct f_Y(y)=... is not valid, it needs for what values of y

[chrome salmon]

2+(1/2) = 5/2

They wrote 3/2

[shadow stream]

oh

I see it

mb!

[shadow stream]

would the range be root(2) and root(3)?

[noble sinew]

here are two examples of how to properly write a PDF

(also in the question you posted (but looks awful because for some reason written in word))

[shadow stream]

why is this true?

for them,
it was 0 < x < 1

[noble sinew]

and what was range of Y?

but I hope you actually check your own book for how method of transformation is given to you in your book, because there are some small variations

[shadow stream]

my professor doesnt have a textbook

he just does problems during lectures and i get lost very often

[shadow stream]

why was the range 1,inf?

[noble sinew]

,w plot 1/x from 0.01 to 1

[ocean seal]

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[shadow stream]

i don't get it

[noble sinew]

1/1=1

[shadow stream]

ik its inf at a certain point

[noble sinew]

1/x decreasing and continious for x in (0,1]

lim x->0^+ of 1/x=inf

so takes any value in [1,inf)

[shadow stream]

oh so its because it starts increasing at 1?

and at anypoint before 1 it's decreasing?

[noble sinew]

no?

range of X is (0,1]

possible values of 1/x is then [1,inf)

as I just showed

[shadow stream]

oh

so for my problem the support would be
(4, 9)?

since g^-1(y) = y^2
and
2^2 and 3^2

[noble sinew]

no?

Range of X is (2,3)

so range of Y=sqrt(X) is (sqrt(2),sqrt(3))

[shadow stream]

ohh i thought we were finding the range with g^-1(y)

okok

Thank you so much!

.close

[lone heart]

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[pearl trench]

heyyy can someone help me pls

[pearl trench]

with this question pls

[modern briar]

help

[pearl trench]

.close

[lone heart]

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[noble sinew]

Means the elements in (A or B) but not in C

Its not division

[lone heart]

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[twin peak]

Hello

[twin peak]

Can anyone explain this please

[lone heart]

@twin peak Has your question been resolved?

[tulip violet]

@twin peak is this the entire question?

[twin peak]

Yes

Nothing more was given

[tulip violet]

May I know what class this is related to

[twin peak]

Grade 8

[tulip violet]

Look at the first 3 set of numbers

Do you see a pattern?

[twin peak]

Yeah

7-1=6

6-1=5

[tulip violet]

Well done!

Yes you’re right the difference between the numbers is 1

How about the second one?

[twin peak]

Thanks! The second one has a difference of 13?

Sorry 8

21-8=13'

[tulip violet]

Well done again

[twin peak]

Woohoo

[tulip violet]

Now how about third one

[twin peak]

I don't

Wait

7 + 14 = 21

[tulip violet]

Close

[twin peak]

21-7=14?

14-7=7?

Maybe this?

[tulip violet]

The answer is 14

But do you understand why it’s 14?

[twin peak]

Because there's a difference of 7?

[tulip violet]

WELL DONE!

[twin peak]

Thanks! I got it now!

[tulip violet]

21-14 = 7

14-7 = 7

[tulip violet]

anytime

[twin peak]

Awesome, thank you!

How do I close the channel?

[tulip violet]

.close

[twin peak]

Thanks

.close

[lone heart]

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@pine igloo Has your question been resolved?

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[loud orchid]

So why Is the answer D and not E?

[placid zinc]

You may be thinking of -(ln(3) + x)

Since they're both in the denominator, they both get a subtraction

[loud orchid]

Ohh okay I see

So anything negative stays in the denominator and positive in the numerator unless there is parentheses

Alright thanks

.close

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[fringe rose]

Hello I need to solve then check for extraneous solutions but I'm stuck

[wary stream]

Your last line is incorrect

[fringe rose]

?

[wary stream]

You wanted to get rid of the $\sqrt{x}$ but you only squared just that term.
From $x = 6 \sqrt{x}$, you tried to square both sides, and only did $x^2 = 6 (\sqrt{x}^2)$ but it should have been $x^2 = (6 \sqrt{x})^2$

[ocean seal]

dldh06

[wary stream]

Hopefully that makes sense

[fringe rose]

.close

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[vernal edge]

I post my questions and no one comes..

[vernal edge]

Is this channel dead

[zealous bloom]

Doesn't look much like a question.

[vernal edge]

I wait 30 mins

[glass lichen]

No, helping is voluntary.

[vernal edge]

Fine I'll retype

[glass lichen]

good idea.

[vernal edge]

I thiught I am shadow banned or smthing

My question is

[glass lichen]

If you were muted you'd have a role...

anyway.

[vernal edge]

How to find area of a circle with given parametric equation r(t)=cos(t) i+ sin(t) j ; 0<=t<=2pi

Basically a unit circle

Using double integrals

I keep getting answer 0

[glass lichen]

double integral with integrand 1

$Area(D)=\iint_D\dd{A}$

[ocean seal]

Mosh

[vernal edge]

Double integral dxdy right

[glass lichen]

whatever order you chose.

[vernal edge]

Yes I am having trouble solving it, I keep getting 0

[glass lichen]

so... post your work

[vernal edge]

What's the mistake here :[

[glass lichen]

the mistake is that that's complete nonsense

[vernal edge]

Yeah I thought so

[glass lichen]

Are you trying to compute with just a regular double integral or Green's Thm?

[vernal edge]

It's outcome from Divergence integral from green theorem,

[glass lichen]

what is the question you're solving verbatum

not an XY problem

[glass lichen]

,rotate

[glass lichen]

Ok so you're doing a regular double integral

[vernal edge]

I just need to know how they gor the pi

[glass lichen]

what's the cartesian eqn of the unit disk?

[vernal edge]

Do you me a circle or a disc disc

Mean*

[glass lichen]

disc

cause a circle has no area, a disk has area

ie the region enclosed by C

[vernal edge]

X^2 + y^2 >= C

No less than equals

[glass lichen]

and not C

unit disc

[vernal edge]

X^2+y^2<=1

[glass lichen]

yes

now what order for dA do you want to use, dxdy or dydx

[vernal edge]

dxdy is okay

[glass lichen]

ok, so what is x in terms of y from the eqn of the circle

[vernal edge]

y=sqrt(1-x^2)

[glass lichen]

+-

[vernal edge]

yes

[glass lichen]

so your x integral has those bounds

what constants is y bounded by?

[vernal edge]

-1 to 1

[glass lichen]

yes

so $\iint_R\dd{A}=\int_{-1}^1\int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}}\dd{x}\dd{y}$

[ocean seal]

Mosh

[vernal edge]

Oh so I don't need that cost and all, it's all Cartesian

[glass lichen]

You'd need the parameterization to do line integrals

however Green's / Stoke's relates line integrals over F to surface integrals over Curl(F)

[vernal edge]

Yes thank you for your help, I'll go solve!! 🙏🙏

[glass lichen]

obviously, you'd rather do this in polar co-ords, but

[vernal edge]

Exactly but I wanted to do this other way

Thanks

.close

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[drifting sigil]

anyone proficient in R, the coding language

[lone heart]

@drifting sigil Has your question been resolved?

[noble sinew]

What about it?

[lone heart]

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[graceful hawk]

I hve a question

[graceful hawk]

wht is, x+x

[dense solstice]

bruh it's unsolvable... u get 1 million usd for this one

[wary stream]

You mean combine like terms?

[graceful hawk]

yeah bro

[dense solstice]

2x

[wary stream]

Don't give answers

[graceful hawk]

2x or xsquare

[raw shard]

obvious question anyways

no point in not giving the answer

[dense solstice]

ok... if anyone ask yuo what's 2+2... u ain't gonna tell him it's 4?

duh

[wary stream]

No, I won't tell the person, I'd say use a calculator or fingers, because basic addition

[graceful hawk]

idiot, i hve a doubt so how can i solve

[kindred hull]

DAMN

[wary stream]

You just add like terms

[graceful hawk]

ok bro

got it

[wary stream]

To combine the terms, they need to have the same base and power

Simple

[graceful hawk]

ok, thnx

[lone heart]

@graceful hawk Has your question been resolved?

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[minor python]

hiiii could someone teach me this i wanted to learn it by my self but i literall could not understand this

[bleak ridge]

Do you know Pythagorean theorem

[minor python]

a little bit

[bleak ridge]

$a^2+b^2=c^2$

[ocean seal]

PapaBread

[minor python]

thannk you

[bleak ridge]

When dealing with points

Their coordinates form a triangle

The x is the base of the triangle and the y is the side of the triangle

And the distance is the hypotenuse

Does that make any sense

[minor python]

a little bit i'm literally sorry

[bleak ridge]

Like this

And the hypotenuse is the length

So rewriting Pythagorean theorem to apply it

[minor python]

ohh i gets it now

thank you

[bleak ridge]

$x^2+y^2=length^2$

[ocean seal]

PapaBread

[bleak ridge]

And notice that that gets the distance to 0

Not the distance between two points

To get the distance between two points you have to subtract the two points

To shift it to 0

[minor python]

is this correct?

[bleak ridge]

Yeah

It first subtracts the two points

Then just plugs it into pythag

It doesn't matter the order you subtract as long as it's the same order for both points too

[minor python]

then how about no.3 i'm literally sorry

[bleak ridge]

You'd just find the distance between each of the three points

And add it all together

[minor python]

thankk you

[minor python]

so no.5 is almost the same with no.2?

[bleak ridge]

Yeah

You're just identifying the points yourself

[minor python]

thankk youu

how to do number 4 last one huhu thank you very much

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[ivory gulch]

can anyone help me with this? i understand linearization

but not sure what im supposed to do ab the inverse part. also worth mentioning, there's an issue in the problem, it's not a one-to-one function

[lone heart]

@ivory gulch Has your question been resolved?

[lone heart]

@ivory gulch Has your question been resolved?

[modest zodiac]

It's one-to-one on a neighborhood of -3, at least, which is enough for what's needed here

Do you know a formula for the derivative of an inverse function ?

[ivory gulch]

no i do not

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[lone heart]

@snow flare Has your question been resolved?

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[void bough]

i have the answer I just wanna know how to get to it

[dark lichen]

For a) it is 180 - 121

[void bough]

waity

is that it

for

[dark lichen]

121 degrees is quad 2

So 180 - 121 is the answer

[void bough]

does that apply to all of them

[dark lichen]

For c it is 360 - 290

The reference angle is 0-90 degrees

Or 0- pi/2

[void bough]

how would you know what to subtract

[dark lichen]

U locate the quadrant in which the angle is located

Then you take the higher number and then subtract

[void bough]

okay

[dark lichen]

90,180,270 or 360

These are the numbers u should subtract the reference angles from

Hope you got it 🙂

[void bough]

I did get it thank you

would it be the same for like acute angles

[dark lichen]

Yup

U do 90 - theta

Tho

[void bough]

what

[dark lichen]

Yea

The acute angle is less than 90

So it’s 90 - theta

[void bough]

ohhhh

tysm

[dark lichen]

Np

[void bough]

.close

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[void bough]

.reopen

[lone heart]

✅

[void bough]

.close

[lone heart]

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[fringe thorn]

#get-advanced-access

[glass lichen]

why did you open 2 channels with that channel....?

[fringe thorn]

I'm new to discordso sometimes I get confused

I'm sorry

[glass lichen]

.close the unneeded channels.

[fringe thorn]

.close

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[edgy cape]

Hello, i got the correct derivatives
dx/dt= 2(4t+9)^-1/2
dy/dx=3e^(1/2x+1)
to get dy/dt I am pretty sure I have to multiply
and the product I get is [6e^(1/2x+1)]/[(4t+9)^1/2]
I then plug in the value t=4 into the original equation to get x=5
and I plug in both t=4 and x=5 into dy/dt but I get a wrong answer
I have attempted this question 3 times
please help

Here is the mark scheme:

[lone heart]

@edgy cape Has your question been resolved?

[edgy cape]

<@&286206848099549185>

[raven swallow]

the idea is correct though, maybe it's wrong significant figures?

[edgy cape]

well so far the answers I got were like 24 and 30

which is wayy off 39.7

so I must have got some value wrong somewhere but I am not sure where

[raven swallow]

Give me sec, I'll solve that too to check the numbers

Yep 39.7

[edgy cape]

what was your x value

[raven swallow]

5

[edgy cape]

same

what was your dy/dt

[raven swallow]

you have put the numbers into the final formula in a cursed way

[raven swallow]

same

[edgy cape]

how

[edgy cape]

Thanks, I think I understand where I went wrong

I was using my calculator incorrectly somewhere haha

[raven swallow]

exactly

that's what happened

[edgy cape]

.close

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[deft stream]

hi guys i need some help with linear and angular motion for my physics assignment

[visual iris]

,,v=rw

[ocean seal]

@visual iris

[visual iris]

Where v is the linear velocity, r is the radius, and w is the angular velocity.

[deft stream]

thx

[lone heart]

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[void bough]

how do i solve this

[buoyant kayak]

by knowing common trig ratios

[void bough]

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[deft agate]

As x --> +(infinity), would the graph go through the y-int = -1? If possible could you show me how the rational graph would look like?

[deft agate]

would the rational graph looks like this?

or do the graph not go through the y-intercept point at all?

Q: When the y-intercept point (0,-1) is the same as the horizontal asymptote ( y= -1). Will the rational graph go through the y-intercept point ?

[lone heart]

@deft agate Has your question been resolved?

[deft agate]

<@&286206848099549185>

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@deft agate Has your question been resolved?

[visual iris]

You might want to check this part again.

[deft agate]

I am not sure what you mean, could you explain to me?

[visual iris]

Huh, that's weird. That contradicts the horizontal asymptote stuff.

Hold on, let me read that again.

[visual iris]

Hmm, because this is saying that if you approach x = -1 from the positive side (from right to left), then f(x) approaches -infinity.

Which would mean this part goes down.

[deft agate]

ohhh i see

this is the first time I seen a rational graph that goes through the hori asymp

[visual iris]

Oh right, you can cross the asymptote. I forgot that was possible. Then that still satisfies the horizontal asymptote thing.

[visual iris]

Yep, I've seen it mentioned before, but this is the first time I've actually interacted with one. :'D

[deft agate]

Thx man! I appreciate your help

.close

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@tawny dawn Has your question been resolved?

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[plucky lynx]

I am having trouble showing the set of all polynomials of degree n with integer coefficients is countable infinite.

[severe sluice]

there are countably infinite coefficients

and there are countably infinite exponents

@plucky lynx also, thanks for being in kanga gang

[plucky lynx]

how is there countable infinite exponents if n is fixed and any polynomials has bases 1, x, ..., x^n?

[severe sluice]

oh

then there can only be finite terms

[small gale]

whats up with names of channels? 🤔
anyway, i had original volume of cylinder, had to find new volume which i did, now i need to found out the percentage increase
original was [pie (r) (r) (h)]
new is 216/125 (original)
how to find percentage increase?

[severe sluice]

the point is that, the cartesian product of a finite number of countably infinite sets is countably finite @severe sluice

(why did i ping myself)

[small gale]

LMAO

[small gale]

am i in right channel btw?

[severe sluice]

no

[plucky lynx]

no, check the available channels

[small gale]

whats the right channel

[severe sluice]

see this section

[small gale]

ight thanks

also whats up with channel names

[severe sluice]

doesn't matter, as long as you know what you're doing

if you want to ask questions, go to math help (available)

if you want to answer questions, go to math help (occupied)

[small gale]

roger that

[severe sluice]

anyways @plucky lynx this is my idea

[plucky lynx]

thanks I try to use that to think of something.

.close

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[alpine sable]

how would i solve this problem?

[buoyant kayak]

expand and combine like terms

[alpine sable]

how do i combine the square root things with the regular numbers?

[buoyant kayak]

you don't

[alpine sable]

how do i expand the problem

[buoyant kayak]

by squaring each of those binomials

[glass lichen]

FOIL.

[alpine sable]

so like (2+2sqr2)(2+2spr2)

[buoyant kayak]

yeh

same with the other one

[alpine sable]

if i do 2sqr2 * 2 would it be 4sqr 2?

[glass lichen]

yes.

[alpine sable]

do i nultiply with the outside numbe

okay

[glass lichen]

you simplify as much as you.

2*2 is 4, so 2*2*sqrt(2) is 4sqrt(2).

[alpine sable]

how do i multiply square root things?

like for 2sqr2 * 2sqr2

8 i think

4+4sqr2+4sqr2+8

8sqr2+12

[buoyant kayak]

now do the same for the other binomial

[alpine sable]

(2-2sqr2)(2-2sqr2)
4-4sqr2-4sqr2+4sqr4
4-8sqr2+4sqr4

8sqr2+12+4-8sqr2+4sqr4
4sqr4+16

8+16

24

imma check my answer rq

WOOO yaa we did it thank you guys

for the help

.close

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[buoyant kayak]

so what's your question

[lone heart]

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[alpine sable]

I’m good on all EXEPT for the division ones

[fleet relic]

which no.?

they all include div

[alpine sable]

what

[fleet relic]

give me which no. ur trying to solve

[alpine sable]

3,2 ,5 , 6

[glass lichen]

Pick one you want to solve / ideally one you have already attempted to solve

[fleet relic]

3. subtract 3 to get -1, then multiply 9 on both sides giving x = -9

do the same steps for all

[alpine sable]

alr ty

[fleet relic]

all you have to do is subtract and multiply

tell me if u got the answer i can check

[alpine sable]

alr

i got x=6

@fleet relic

vm

[fleet relic]

for which question

@alpine sable

[alpine sable]

i got for number 3

[fleet relic]

no i did #3 for you

x=-9

here ill do umber 2

1 + r/9 =0
subtract 1 ffrom both sides
(1-1) + r/9 = 0-1
r/9 = -1
multiply 9 from both sides
9(r/9) = -1(9)
r= -9

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[soft folio]

hey guys i have a math question

[buoyant kayak]

well that's strange

[soft folio]

its on Analytic Geometry Problems

how do you find the shortest distance if you have the equation of the line and a point

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[tall wing]

.close

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[olive inlet]

Help

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[alpine sable]

Could someone help me out, I don't understand the language of my book.

[alpine sable]

"What is the projection on the line of equation x = y of the point (3,-1)"

I have no idea where the transformation matrix is coming from

[vale wigeon]

have you worked with linear transformations before

[alpine sable]

Why is it not just (3,3)?

Yes, I don't understand what the sentence is saying though

[vale wigeon]

which sentence?

[alpine sable]

"What is the projection on the line of equation x = y of the point (3,-1)"

[vale wigeon]

you want to project the point (3, -1) orthogonally onto the line y=x

[alpine sable]

Ahhh

Thanks very much

lol silly question I apologize

[vale wigeon]

like this

[alpine sable]

Yeah I thought they wanted the perpendicular dropped from the line to the point. I appreciate the help

.close

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[waxen harbor]

Anyone know why the answer to this is E?

[gusty yew]

Pretty sure it's d,

[vale wigeon]

who's telling you it's e?

[waxen harbor]

dang my teacher wrote E

can you explain why its D? im confused on spanning sets

[vale wigeon]

C is literally the standard basis of R^2

B is a superset of C and hence spans at least as much as C does

A spans only the line y=x, as its span is equal to span{(1,1)}

[waxen harbor]

Hmm okay. So since B is only different from C by adding 0,0 then that means it is super set

I'm still a little confused on how you know A only spans y=x

[vale wigeon]

(-1/2, -1/2) is a scalar multiple of (1, 1) so it can be removed from A without affecting its span

[waxen harbor]

Okay

Thank you

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[grave urchin]

Is this right?

[vale wigeon]

there is a hole

other than that yes this is right

[grave urchin]

would it be (-3,-2)? I factored the top and bottom and simplified it as f(x)=-2

i got -3 from finding x from the factor (x+3)

[vale wigeon]

yes, that's where the hole is.

[grave urchin]

if theres a hole there shouldn't there also be a vertical asymptote there?

[vale wigeon]

no

hole ≠ vertical asymptote

a hole is a missing point that can be 'filled in'

[grave urchin]

ah i undertand now ty

.close

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[quartz viper]

.reopen

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[manic prism]

Can anyone explain to me about anti symmetry for part a ??

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[vital timber]

if i wrote this out long form

[vital timber]

would it be multiplied?

or added?

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[alpine sable]

can someone push me in the right direction please i know the formula for both the length and halfway point but need to find the coordinates from this question

[alpine sable]

i just need pushing in the right direction

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@alpine sable Has your question been resolved?

[smoky cloud]

u got a and b ?

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[alpine sable]

Using the graphs, find the solutions of the equation 2x^2 = x^3.
Image

[noble sinew]

just look where they intersect

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[alpine sable]

show that the given equation has no integer solutions. x3 + 3x + 3 = 0.

[vocal horizon]

x3 is x.x.x?

[ocean seal]

......................

[glad tide]

id assume x^3, since x3 does indeed have an interger solution

[alpine sable]

x3 + 3x + 3 = 0

[glad tide]

what is x3

[alpine sable]

equation

[glad tide]

so it's x^3?

or 3x

[alpine sable]

3x

[glad tide]

ah so 3x + 3x + 3

[ocean seal]

AlphaZero

AlphaZero

AlphaZero

AlphaZero

[alpine sable]

thank you👍

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[manic prism]

Can someone help me for an explanation about anti symmetry and how to prove it for part a ??

[vague coral]

x = y^2 isnt the same as x^2 = y
hence the anti-symmetry

@manic prism

[placid zinc]

Anti-symmetry says that if x = y², then it MUST NOT be true that x² = y

[vague coral]

symmetry means that for a binary relation between two elements, if you have a = b, b=a should mean the same

[placid zinc]

So note that a is almost anti-symmetric, but isn't quite. (1,1) is a case that breaks anti-symmetry

[manic prism]

I need an simple explanation if you like to

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@manic prism Has your question been resolved?

[placid zinc]

Oh wait, whoops I am mistaken. (1,1) is allowed in an anti-symmetric relation

As 1 and 1 are the same, of course

I forgot that part of the def, my bad

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@manic prism Has your question been resolved?

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[formal grove]

anyone know answer to this

[lone heart]

@formal grove Has your question been resolved?

[lost steeple]

I believe you just plug the values in

Find the values

Then use slope formula to calculate it

(g(3)-g(-4))/3-(-4)

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[median mauve]

c is even how to proof it work

[alpine sable]

note first that such a consecutive sum is always a difference of triangular numbers

[median mauve]

wdym

[oak chasm]

A triangular number is the sum of all integers from 1 to some n.

The sum of consecutive integers is a sum of all integers from some m to some n.

So, if you take the triangular number for 1 through n and subtract the triangular number for 1 through m - 1, that gives you the sum of consecutive integers from m to n.

[median mauve]

huh

can u make an example

[oak chasm]

Well, let's say we want the sum from 6 to 8.

That's 6 + 7 + 8.

Well, that's the same as this:

(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8) - (1 + 2 + 3 + 4 + 5)

[median mauve]

oh

ok

but how this connect the problem'

(⊙_⊙)？

[oak chasm]

OK, so do you know from algebra the formula for the sum of the first n integers?

[median mauve]

(a1+an)*n/2

[oak chasm]

OK, yes n(n + 1)/2.

So, we need to subtract the sum from 1 to m - 1 from the sum from 1 to n to get the sum from m to n.

n(n + 1)/2 - (m - 1)(m - 1 + 1)/2

So, (n(n + 1) - m(m - 1))/2.

[median mauve]

ok

[oak chasm]

That's the sum of the consecutive integers from m to n.

[median mauve]

ok

but how can that prove c can be sum of 2 or more consecutive int

when c is even case

[oak chasm]

So, c = (n(n + 1) - m(m - 1))/2

Where n > m because there's at least two integers in the sum.

2c = n(n + 1) - m(m - 1)

[median mauve]

?.

[oak chasm]

Replace m with n - a, then factor.

What do you get?

[median mauve]

n(n+1)-(n-a)(n-a-1)

[oak chasm]

OK, now expand and then factor.

[median mauve]

2n+2an-a^2-a

2n(a+1)-a(a+1)

(2n-a)(a+1)

[oak chasm]

Good.

Now a can't be less than 1 because that's how much less the first number is than the second and they're different numbers.

So, a ≥ 1.

So, let's look at this for a moment.

2n - a is the form of what?

Sorry, corrected it. a ≥ 1.

[median mauve]

2n-a?

[oak chasm]

Yes, that looks like the form of what?

[median mauve]

idk

[oak chasm]

Well, from yesterday, we have 2n + 0 as the form of even numbers and 2n + 1 for odds, but it really works for 2n ± whatever.

If the number added or subtracted is even, it's the form of an even number. If the number added or subtracted is odd, it's an odd number.

Does that make sense?

[median mauve]

yes

[oak chasm]

OK, so we have:

2c = (2n - a)(a + 1)

Let's say a is odd.

Well, the 2n - a part is odd.

The a + 1 part is even.

And vice versa. If a is even, 2n - a is even and a + 1 is odd.

There's a 2 on the c.

So, we know by whether a is odd or not which factor on the right gets that 2.

Let's put it in the a + 1 factor.

Let's say a + 1 = 2 to make things simple.

[median mauve]

ok

so a = 1

[oak chasm]

So, c = 2n - 1

Well, we can get any odd number we want with that.

If we want 3, we just set n to 2.

Etc.

So, we've proven you can get any odd number.

What about even numbers?

[median mauve]

let a = even

[oak chasm]

Well, you could do that, but let's do this instead.

Make a + 1 = 4

Why?

Well, because now we have:

2c = 4(2n - 3)
c = 2(2n - 3)

We can get any odd number with 2n - 3.

And then we multiply it by 2 to get all the even numbers that have 2¹ in their prime factorization.

Does that make sense?

[median mauve]

yes

[oak chasm]

@median mauve So, let's make a + 1 = 8 (2³).

Then we'll be able to get even numbers that have 2² in their prime factorizations.

If we make a + 1 = 2ⁿ, we can get all even numbers with 2ⁿ⁻¹ in their prime factorizations.

Does that make sense?

[median mauve]

yees

[oak chasm]

But there's a limitation.

It's not really all even numbers.

It's all even numbers that also have odd primes in them.

So, any number that's not of the form 2ⁿ alone.

Which is what we're proving.

If we want to prove it for a particular number that's not a power of 2.

We just get the power of 2 in the prime factorization.

Set a + 1 to 2ⁿ⁺¹.

Set the other factor to the odd part of the integer.

And then we got it.

Does that make sense?

[median mauve]

yes

so let me try to write a conclution

We can write the sum of consecutive numbers from m to n into the form of n(n + 1)/2 - (m - 1)(m - 1 + 1)/2, so we can simplify to (n(n + 1) - m(m - 1))/2 which is the sum of the consecutive integers from m to n. By replacing m with n - a, and factoring we get it is equal to (2n-a)(a+1). Hence we have 2c = (2n - a)(a + 1). If we make a + 1 = 2ⁿ, we can get all even numbers with 2ⁿ⁻¹ in their prime factorizations. Thus, all even number expect 2ⁿ can be written in the form of sum of consecutive numbers.

how about this

[oak chasm]

Well, it needs more detail.

One thing, though, you have n representing two things.

In one, it's the upper number in the sequence of integers.

In the other, it's the power of 2 in the prime factorization.

So, that can be confusing.

[median mauve]

how did you make it smaller

[oak chasm]

The superscript ⁿ?

[median mauve]

yes

[oak chasm]

Oh, I went to some Wikipedia pages with Unicode (the character set that we use nowadays with different languages and emojis and such) math symbols.

Then I went into my iPhone's text replacement settings and made things I can type to get those math symbols.

Like I type ^^n to get ⁿ.

[median mauve]

nice

[oak chasm]

But simpler than that is to just search for unicode superscript and copy and paste from a page that has them.

Unicode is the keyword for special symbols.

[median mauve]

We can write the sum of consecutive numbers from m to n into the form of n(n + 1)/2 - (m - 1)(m - 1 + 1)/2, so we can simplify to (n(n + 1) - m(m - 1))/2 which is the sum of the consecutive integers from m to n. By replacing m with n - a, and factoring we get it is equal to (2n - a)(a + 1). Hence we have 2c = (2n - a)(a + 1). If we make a + 1 = 2ᵏ, we can get all even numbers with 2ᵏ⁻¹ in their prime factorizations. Thus, all even number expect 2ᵏ can be written in the form of sum of consecutive numbers.

[oak chasm]

OK, it needs more details.

Like you should point out that 2n - a would be odd.

And that you can get any odd natural number from it.

Why?

Because without knowing that, a + 1 is a power of 2 and 2n - a might be a power of 2.

And then you can't get 6, for example.

For 6, you need a power of 2 times 3.

So you need to be able to get odd numbers as one of the factors.

So you need to show that you can.

[median mauve]

I need to show 2n-ais odd?

[oak chasm]

Yes, and not only that.

Let's say that I want to get 6 to work.

a + 1 can be a power of 2 that works.

We need it to be 2² and we can get that.

We just set a.

What do we need 2n - a to be?

[median mauve]

2n - 4

[oak chasm]

Right, but if you show that 2n - a is odd, it shows that it's odd, but it doesn't show that 2n - a can be 3 in particular.

Well, not right.

You need 2n - a to be 3.

Do you see why?

[median mauve]

because consecutive?

[oak chasm]

No, because 6 = 2·3.

[median mauve]

oh

[oak chasm]

The even part is handled by a + 1.

The odd part by 2n - a.

So, 2n - a needs to be 3.

If you prove 2n - a is odd, that's not enough.

It has to be 3 in particular.

Different numbers will have different odd parts.

You need to be able to get any of them that might show up.

So, 2n - a needs to be able to achieve ALL odd naturals.

Can it?

Like if my odd part is 1003094031, can 2n - a get me that?

[median mauve]

yes？

[oak chasm]

Why?

[median mauve]

hmm

i confuse

[oak chasm]

What are you confused on?

[median mauve]

base on my understanding

• a+1 is the even part which is $2^n$
• 2n-a is the odd part which is c/(a+1)

[ocean seal]

LittleMouse

[oak chasm]

OK, but what are you confused about?

[median mauve]

here

[oak chasm]

Oh, OK.

Well, what is the form of an odd number?

[median mauve]

2n+1

[oak chasm]

Right, but that's not all.

2n ± an odd number.

[median mauve]

yes

[oak chasm]

2n - 11 also works.

[median mauve]

ok

[oak chasm]

You can convert that into 2n + 1.

2n - a
2n - (2x + 1)
2n - 2x - 1
2n - 2x - 2 + 1
2(n - x - 1) + 1

Now you have it in a more standard form if you replace the odd number a with its form of 2x + 1.

[median mauve]

ok

[oak chasm]

We don't need to get n - x - 1 to be 0.

Because we don't need 1 to work as the odd part.

Why not?

[median mauve]

cuz it will be same as only even part?

[oak chasm]

Yes, and the even part is a power of two.

And we don't care about those.

If the whole integer is a power of 2, we can ignore it.

So we need n - x - 1 to be able to give us any integer greater than 0.

If it can, we can get any useful odd number we want.

If we need 3 for the odd part, we just make that 1.

If we need 1003094031, we get

,calc (1003094031 - 1)/2

[ocean seal]

Result:

5.01547015e+8

[oak chasm]

Whatever that is.

And so on.

So, the question is: can n - x - 1 give us any integer greater than 0?

[median mauve]

yes

[oak chasm]

How?

[median mauve]

why not

[oak chasm]

What do you mean?

[median mauve]

aaa i feel so confuse

so what is x

[oak chasm]

Do you want to take a break?

[median mauve]

I can do it

[oak chasm]

OK, so we had 2n - a for the odd factor.

We want to get that in the form 2j + 1 for some j.

[median mauve]

ok

[oak chasm]

a is odd.

So, a = 2x + 1 for some x.

2n - a
2n - (2x + 1)
2n - 2x - 1
2n - 2x - 2 + 1
2(n - x - 1) + 1

Now we have it in the form 2j + 1 for some j.

[median mauve]

ok

[oak chasm]

We need 2(n - x - 1) + 1 to give us any odd number greater than 1.

So, what does that mean n - x - 1 has to be?

2(n - x - 1) + 1 > 1

Solve for n - x - 1.

No.

[median mauve]

2(n - x - 1) + 1 > 1
2(n - x - 1) > 0
(n - x - 1) > 0

[oak chasm]

Right.

And further, n - x > 1.

a is already set by the even part of the number.

So x is already set because it's based on a.

Does that make sense?

[median mauve]

yes

[oak chasm]

So what is the restriction on x?

What inequality can you write for it?

[median mauve]

restriction

[oak chasm]

Yes.

Can x be -10039434?

[median mauve]

no

[oak chasm]

OK, so there's a lower bound.

[median mauve]

a = 2x + 1

x>=0?

[oak chasm]

Why?

[median mauve]

a >= 1

?

[oak chasm]

OK, good.

So, n - x > 1.

Can x increase the number on the left?

[median mauve]

increase the number on the left?

[oak chasm]

Yes. We have n - x > 1. n - x is on the left.

[median mauve]

oh

as x decrease n-x increase

[oak chasm]

OK. Can x increase n - x above n?

[median mauve]

no

[oak chasm]

Right, so - x is going to be nonpositive.

What's the restriction on n?

[median mauve]

n>x+1

[oak chasm]

No, think back to the problem.

What does n represent?

[median mauve]

consective interger

[oak chasm]

Yes, the upper one.

So, these are two consecutive positive integers.

So, n can go as high as it wants.

How low can it go?

[median mauve]

m+1

[oak chasm]

No, as an integer with no variables, how low can n go?

[median mauve]

0?

[oak chasm]

OK, let's do some examples.

1, 2, and 3.

That's at least two positive consecutive integers.

Can the highest one get lower than 3?

[median mauve]

no

[oak chasm]

Why not?

[median mauve]

other wise m is going 0?

[oak chasm]

Can you show me why?

[median mauve]

if highest be 2

and it need be at least 2 consecetive

then it is being 0

[oak chasm]

OK, make some example sequences of at least two consecutive positive integers.

[median mauve]

123?

[oak chasm]

What else?

[median mauve]

234

345

[oak chasm]

What about sequences of other lengths?

[median mauve]

23

45

67

[oak chasm]

OK.

Also, 4, 5, 6, 7 and so on.

Now what's the shortest sequence you can make?

[median mauve]

2

[oak chasm]

What's the lowest you can get the numbers to go?

[median mauve]

1

[oak chasm]

Which sequence is that?

[median mauve]

12

[oak chasm]

Right, and 2 is the lowest n can get.

Do you see why?

[median mauve]

yes

[oak chasm]

OK, so n - x > 1.

n ≥ 2.

x ≥ 0.

[median mauve]

ok

[oak chasm]

So, let's say x is 30.

n - x has to be able to give us anything greater than 1.

How can we get n to do that for us?

[median mauve]

32