The Grove

Deusovi

00:00

rSₙ = [ ra₁ + r²a₁ + ... + rⁿ-¹a₁ ] + rⁿa₁

Prince North Læraðr

(Hehe, those induction proof tricks coming in handy)

Ohhh

What

(Okay laptop. anywhom)

Deusovi

same thing as before, i just bolded stuff and put some suggestive brackets around some of the terms

the bolded part here is just Sₙ, but without a₁

Prince North Læraðr

Right

Deusovi

in other words, rSₙ = [ Sₙ - a₁ ] + rⁿa₁

Prince North Læraðr

Huh

Deusovi

00:01

does that make sense?

Prince North Læraðr

Yup

Deusovi

and now this is a nice, simple equation - we can just solve for Sₙ!

Prince North Læraðr

Eek

Deusovi

it's not as complicated as it looks, don't worry

Prince North Læraðr

hm, if you say so :P

Deusovi

00:03

just do the usual thing you do - get all the terms with "X" on one side, the terms without "X" on the other, and then factor out "X"

except instead of x, it's Sₙ

Prince North Læraðr

Okay, I'm just gonna substitute S_n for x so it's easier on my brain :P

Deusovi

sure!

Prince North Læraðr

(algebra-ing)

Is [ Sₙ - a₁ ] + rⁿa₁ = Sₙ - a₁ + rⁿa₁ ?

Deusovi

yep

Prince North Læraðr

Okay, so the brackets were there just for clarification

Ah, I see

Deusovi

00:07

yep!

Prince North Læraðr

(if x=s_n then the following ensues)

(also pretend a=a_1, just to lesson the amount of symbols)

rx=x-a+(r^n)a
rx-x=-a+(r^n)a
x(r-1)=-a+(r^n)a
x=(-a+(r^n)a)/(r-1)
x=a(-1+r^n)/(r-1)

Hm, that's weird

It's 1-r^n, not -1+r^n

scratches head while looking back at work

bobble

your denominator is r-1

Prince North Læraðr

Oh yeah that doesn't match the equation either

Oh, is it just because of how I moved the variables?

Seems to be so

yup

Well I won't repaste everything I just re-did but I did get the same answer as that equation above

How is a((r^n)-1)/(r-1) equal to a(1-(r^n))/(1-r)?

Eh, you know what, question for another day, I have way more work to get done

bobble

multiply top and bottom by -1

Prince North Læraðr

That gives you -a though

Oh, before you take a out, duh

Or not even

Nvm yeah that makes sense

Prince North Læraðr

00:39

@Deusovi I need some help

bobble

\o

Prince North Læraðr

Okay so

Deusovi

hm?

Prince North Læraðr

I'm not sure how to go about this one

I know it's

Deusovi

you'll want to get the left side to a closed-form expression (something without the ∑ sign in it)

Prince North Læraðr

00:41

Hm, so like....

(28-4)+(28-4*2)+...(28-4*n)=72?

bobble

lookit your formula slide

Deusovi

you don't want ...s either

Prince North Læraðr

Idk though like what the a_n term is though

Hng

Deusovi

aᵢ = 28-4i. they're just using i as an index, because n is being used for something else

Prince North Læraðr

(processing)

Okay

Wait why is a_i=28-4i?

Deusovi

00:44

well, it doesn't really make sense to talk about a_whatever in this problem, because that's not defined anywhere

that's not a variable that exists

Prince North Læraðr

Oh

Deusovi

@PrinceNorthLæraðr i assumed you wanted to know the general term of the thing being summed, which is what i gave you

but i don't know what exactly you were asking for, and i may have misinterpreted you

Prince North Læraðr

sorry this is just a bit confusing. Hold on, trying to process this

What do you mean by the general term?

Deusovi

i mean "the formula describing the thing being summed"

"general term" is the opposite of "specific term". one specific term is 28-4. the next specific term is 28-8. the next is 28-12. the way to describe all of these in general is "28-4i"

Prince North Læraðr

Right....

But we're also looking for the sum?

Deusovi

00:47

yes, that is true?

the sum of multiple things that can all be described as "28-4i" is indeed occurring in this problem

Prince North Læraðr

Sorry, I guess I'm just having trouble conceptualizing what a general term would look like

So like the common difference here is 4

Deusovi

look, just ignore "general term"

Prince North Læraðr

Okay

Deusovi

"28-4i" is the general term that i was referring to

that's all. i was just giving you that, because that's what i thought you were asking for

Prince North Læraðr

(Sorry if I'm frustating you)

Deusovi

00:49

you're fine, you're fine

Prince North Læraðr

I was referring to the equations I had

Deusovi

ahhhh okay

Prince North Læraðr

Yeah, that may have been the confusion

Deusovi

what you actually want is what's called a "closed form" - a way to write the thing without ∑ or ...

once you have that, you can do the algebra

Prince North Læraðr

Hm, okay

Deusovi

00:51

so you have ∑ᵢ₌₁ⁿ (28-4i) = 72. i doubt you have a formula to transform ∑ᵢ₌₁ⁿ (28-4i) into something nice. what sum rules do you know?

Prince North Læraðr

S_n=1/2n(a_1+a_n)

Deusovi

hold on a sec

Prince North Læraðr

Sure

Deusovi

that's not always true. what is the precondition for that?

Prince North Læraðr

Oh, it's when you're adding it, isn't it

When the common difference is positive?

Deusovi

00:53

ah, you're assuming a common difference exists

Prince North Læraðr

Ohhh

Deusovi

the precondition is "the things being summed are an arithmetic sequence"

Prince North Læraðr

This is not an arithmetic sequence?

Deusovi

so, you have recognized that that applies here, but you didn't say it so i wasn't sure

Prince North Læraðr

Ah, okay

Deusovi

00:54

so, what does that become here? what is a₁, and what is aₙ?

Prince North Læraðr

So we have a_1 term, that's 28-4 or 24

Hm, not sure about a_n

Is it 28-4i?

Wait...

(thinking)

I'm not sure

Deusovi

can you explain how you got a₁?

what action did you take?

Prince North Læraðr

Well, since the summation thingy above said that "i" term starts at one, the first term would be 28-4(1), which is 24

Deusovi

okay, and you got 28-4(1) how? you took an expression from somewhere, and did something to it

Prince North Læraðr

I took the expression from inside the summation thingy (28-4i)

And plugged in 1, the first term, for a_1

Deusovi

00:59

okay, so do that

take the expression, and plug in n

Prince North Læraðr

Hm, okay

28-4n

And that's my a_n term

Deusovi

yep

remember, n is just a stand-in for a number. you would know what to do if someone gave you a number (if i asked "what's the millionth term?", say) - so just do that, but with n instead!

Prince North Læraðr

(hold on, I gotta walk my dog, but it seems like I need to factor)

Prince North Læraðr

01:12

(back)

@Deusovi Um am I supposed to be doing quadratic formula?

Deusovi

have you found a quadratic equation? if so, the quadratic formula seems to be reasonable

Prince North Læraðr

well it seems to be (n^2)-52n+144

Shouldn't n be a natural number?

Deusovi

i would certainly hope so

i haven't done the math myself but getting a quadratic equation sounds reasonable

Prince North Læraðr

scratches head

Deusovi

@PrinceNorthLæraðr what exactly seems to be that

Prince North Læraðr

01:19

-52+-sqtr(2128)/2

Deusovi

no, i mean

you said "it seems to be..."

what is "it"

Prince North Læraðr

Oh. Zero

Deusovi

did you multiply both sides by 2 to clear fractions?

Prince North Læraðr

Yeah. Here's my process

Oh it's 4n

smh

I've been doing n(52-n) not n(52-4n)

Deusovi

ah

that would explain things

Prince North Læraðr

01:22

Let me just take a 4 out now

Welp some more oddities

0=4((n^2)-13n+36) so 0=4(n-9)(n-4)

I would assume n=4, but like how would one know?

Deusovi

4(n-9)(n-4) = 0

you have three things that multiply to get 0, so at least one of them must be 0

therefore either 4, n-9, or n-4 is zero

Prince North Læraðr

Hehe, well 4 can't be zero

Deusovi

i hope not! that would break all of math as we know it

which would probably cause a few problems

2

Prince North Læraðr

Deusovi: "That would break all of math as we know it. Which would cause a few problems"

Anyhow

So it's either n-9 or n-4

Deusovi

right, so either n-9=0 or n-4=0

Prince North Læraðr

01:26

Right

Deusovi

and that's your answer

Prince North Læraðr

Oh

Deusovi

both n=4 and n=9 work

Prince North Læraðr

O_o how??

Deusovi

the thing you're summing starts out positive, but then when n gets bigger it dips into the negative

Prince North Læraðr

01:27

Right

Oh, does it start cancelling terms with the positive values?

Deusovi

so you go above 72 after a few terms, then the sum drops back downwards (as the numbers you add become negative) and it hits 72 again

Prince North Læraðr

ohhhhh

That's very odd

Deusovi

in fact, you can actually see when that happens - you're adding up 28-4i, right?

so the terms you're adding go like:
24, 20, 16, 12, 8, 4, 0, -4, -8, -12...

Prince North Læraðr

Ohhh I see

Deusovi

that means the sum will stay at its peak for one... time-unit-thingy, and then drop back down, undoing the old terms one-by-one

Prince North Læraðr

01:33

Right

Deusovi

hopefully that made sense! anything else?

Prince North Læraðr

We'll see :) I'm just going through more problems right now

Deusovi

alright!

Prince North Læraðr

I'll ping you if I need more help. Thanks!

Deusovi

alright, sounds good! happy to help

Prince North Læraðr

20:35

Hehe @Sciborg thought I'd ping you here about his sword. I believe I'm too far away

I'm 80 feet from him, and will be moving even further back and sniping him with arrows instead

Sciborg

Ah, okay. i wasn't sure if you were gonna do melee or not

Prince North Læraðr

Nah. Makes it harder to hit me if I'm further away

If I'm melee, then I figured he just has disadvantage trying to hit me, but if I'm far away, not only does he have disadvantage but he also has to locate where I'm at

I give my location away by attacking, but don't think I'll be just staying in one place

Btw how are you going to have Varlan guess my location (if he even wants to bother fighting me Ig)

Prince North Læraðr

21:16

What spell is this

in PSE D&D Chatroom, Apr 22 at 22:59, by Sciborg

"I have a spell for that," Raji says, and does a dismissive flick of his hand. A tiny, floating eye appears, and hovers alongside the wagon. "We'll be fine. It'll make a noise if it sees anyone coming."

This isn't Scrying, is it?

Ah, alarm

Sciborg

Alarm, yeah

Prince North Læraðr

I was like

THIS MAN HAS SCRYING

bobble

dear GOODNESS, the page with instructions on how to apply for housing is SO LONG

Prince North Læraðr

Oof :(

bobble

there are too many steps and forms and aggggggggh

Prince North Læraðr

21:26

Yikes

(uh-oh, why do I have a bad feeling about Kith and Johnny?)

87

Q: Do Cantrips use your character level or class level?

Cantrips seem to be the only type of spell that cares only about your class level. However, in the multiclassing section they are not mentioned, does this mean that cantrips use your character level, instead of your class level? Does this mean a Warlock 2 / Fighter 15 can cast an Eldritch Blast ...

Oooooooh

So all of my cantrips get a nice buff next level

Oh, let me guess - they're going to the Hearth to see Effren or Mikael? :P

Prince North Læraðr

21:54

O_o

Dimension door is a 4th level spell

That means if Mikael is a pretty damn powerful wizard

@Deusovi Are you around?

Deusovi

yeah, what's up?

Prince North Læraðr

Okay, so I had some more problems for infinite sums I was kind of confused with

This question is the first one I'm a bit stuck on

Deusovi

alright, where does the work stop making sense

do you not understand what's going on at the start? partway through? the last line?

Prince North Læraðr

I'm confused on why we're dividing everything, and then also why there's suddenly a subtraction sign

Deusovi

you know that the sum of a geometric series is always "a/(1-r)"

Prince North Læraðr

22:07

Wait, it's always 1 /(1-r)?

Deusovi

whoops, right, 1 iff the first term is 1

Prince North Læraðr

Okay, I was slightly confused :P

Deusovi

so your goal is to get "2/(2+x²)" to look like "something/(1 - something)"

Prince North Læraðr

Right

So, that's why she divided everything by two?

Deusovi

first, you want that 2 on the bottom to become 1, so

yeah exactly

Prince North Læraðr

22:08

That's allowed? Huh

I mean

I guess I don't see why not

Deusovi

you're multiplying by (1/2)/(1/2)

Prince North Læraðr

I thought there was some weird partial fraction decomposition involved

Ah

Deusovi

which is just 1, and multiplying by 1 is always allowed!

Apr 14 at 22:35, by Deusovi

lots of clever tricks for solving problems are just "multiplying by 1" or "adding 0"

Prince North Læraðr

So we have

1/[1+(x^2/2)]

Oh, and then she just

Deusovi

not quite, it's x²/2 - remember to distribute the bottom 1/2

Prince North Læraðr

22:10

Made (x^2)/2 into a negative term

That's a bit strange workaround

Deusovi

yep, exactly!

Prince North Læraðr

It's like the opposite of simplification :P

Deusovi

you need that minus sign, so you just make it minus a negative, and your common ratio is negative

heh yeah

but then you can shove all of the mess you've just made into the variable r

Prince North Læraðr

right

Okay, and then everything else makes sense

Okay, here's the next one

Why is it 3/2-1 <x< 1+3/2 ?

Deusovi

the common ratio is (2/3)·(2x+1)

Prince North Læraðr

22:14

Right

Deusovi

for the series to converge, you need |r|<1

Prince North Læraðr

Mhm

Deusovi

so that gives -1 < (2/3)·(2x+1) < 1

Prince North Læraðr

Yesssir

Deusovi

and that's multiplying by 3/2, not adding it - they miswrote it on the left side there

Prince North Læraðr

22:15

Ohhh

Deusovi

yeah that's pretty annoying

Prince North Læraðr

Yeah

Okay, well everything else makes sense then

I was wondering why the heck you were dividing by 3/2

Yay!

I solved a tricky looking problem on my own!

Deusovi

nice!

Prince North Læraðr

Though, admittedly, I solved it very differently from my teacher

I started by turning the sum into an equation, so 1/(1-(cos(theta))^2))=5

(since the first term is zero, cos(theta)^2(0) is 1)

And then saw that huh, 1-cos(theta)^2 is just sin(theta)^2

So 1/sin^2=5 (theta implied)

which means sin^2=1/5

And then used sin^2+cos^2=1 to find cos^2

And cos(2theta) is just cos^2-sin^2

I did some other work before realizing that it wasn't nearly as complicated as I thought

(personally, I think my teacher did way more extraneous work to get to the answer :P)

Deusovi

yeah, they wrote it out a bit more thoroughly

Prince North Læraðr

22:29

yeah

Deusovi

and your variable substitution, solving for sin²θ, could make things a bit easier (and gives you slightly more freedom near the end)

other than that, seems like roughly the same thing? but more importantly, your logic made perfect sense

Prince North Læraðr

YSee

Ugh, this unit is forcing me to remember stuff from Algebra 1 that I just burned off my memory

Like

vertex form

Okay so again, I started a little differently

I said that since a2=a1*r, and a2=17, 17=a1*r, or a1=17/r

(17/r)/(r-1) is just 17/(1-r)r or 17/r^2-r

Now I'm slightly confused on what to do next

Deusovi

you want to make 17/(r²-r) as small as possible, right?

to make a small fraction, you want a small numerator and a large denominator. you can't change the top of this fraction, so you want to make the bottom as big as possible

Prince North Læraðr

Okay

Deusovi

in other words, you want to maximize r²-r

Prince North Læraðr

22:43

Okay. And that's what we use vertex form for

Deusovi

right, since that's a quadratic

Prince North Læraðr

And I completely forgot everything about vertex form -_-;

well out zeroes are between o and 1, I guess that makes sense

because r(r-1)=0 produces r=0 and r=1

Oh, and the b is the coefficient of the b term of the quadratic r²-r? So -1?

Deusovi

yeah

Prince North Læraðr

Why did she put -1 on the top instead of -(-1)?

Deusovi

@PrinceNorthLæraðr you can do it this way, too - what do you know about how the maximum relates to the zeroes of a quadratic?

Prince North Læraðr

22:46

@Deusovi It's in the middle of the two zeroes

Oh, so you can find

The average?

Deusovi

@PrinceNorthLæraðr oh, because you got it backwards here

Prince North Læraðr

@Deusovi ah, mb

No, you find the halfway point between the two points

Deusovi

so that should be r-r², and just carry the negative sign through

anyway

yep, you can just find the midpoint of the two x-values, and that's where your vertex is!

Prince North Læraðr

Midpoint formula was what again (googles)

Oh it is just the average :P

Deusovi

yes

Prince North Læraðr

22:58

Hm, I'm solving a lot of these concepts in a way that makes sense but not the way my teacher did it and I'm a bit worried

So this one

We have a+ar+ar^2 + etc =7

But we also have ar+(ar^3)+(ar^5)+etc=3

So my immediate thought process was to take an r out which makes r(a+(ar^2)+(ar^4)+etc)=3

And my first brain process was, "Well, if all of the odd terms equal three, and I just manipulated the equation, giving myself all the even terms, that means that the even terms must be four! And then I was like r has to be a number that turns the 4 of all the even terms into three

As in

a+(ar^2)+(ar^4)+etc=4, but r(a+(ar^2)+(ar^4)+etc)=3

And then I messed up basic math and solved r as 4/3 and I was like hm

4/3 is greater than one, so I guess my assumption was wrong

method two:

r(a+(ar^2)+(ar^4)+etc)=3
a+(ar^2)+(ar^4)+etc=3/r
a+(ar^2)+(ar^4)+etc+ ar+(ar^3)+(ar^5)+etc=7
That means 3+3/r=7

Math the rest and then you get 3/4

And then I realized back that my assumption of the even terms adding up to four was indeed correct and smacked my forehead

But I like method two better anyways, I wasn't confident that the even terms actually added up to four

(Thanks math induction for teaching me short cuts!)

Deusovi

23:21

yep, your version looks good to me!

both of them do

Prince North Læraðr

Yeah

Idk it's just like

I don't know if I should be doing her way

Deusovi

honestly i think the way you did it with "all the even terms add to 4, and if you multiply the even terms by r you get the odd terms" is how i would do it

your logic makes sense and is mathematically sound, and you got the correct answer. there's no need to do things in the exact same way your teacher did - you're fine

Prince North Læraðr

Okay

It just worries me for the test, that's all

Deusovi

you're learning how to do more complicated problems where there are many different approaches - it's not just "here's a bunch of formulas, here's where you use those formulas" anymore

as long as your approach makes sense and leads you to the right answer, you're 100% okay

Prince North Læraðr

Okay

Prince North Læraðr

23:57

Hehe

It's really weird that

-2<-x<2 is equal to -2<x<2

Because when you divide out the -1 it becomes

2>x>-2 which is literally -2<x<2

bobble

or you could just take the first thing to mean "|x| < 2"

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