The Last Post Wins!!!!!

[Vyzkel He/Him]

3 minutes ago, Through The Living Glass said:

No you're a father

But I could’ve been a mother

[Through The Living Glass She/They]

10 minutes ago, Ookla the Invyzable said:

But I could’ve been a mother

True. I am sorry for taking your motherhood away from you.

[Vyzkel He/Him]

10 minutes ago, Through The Living Glass said:

True. I am sorry for taking your motherhood away from you.

It’s okay, I’ve heard it’s a hard job.

[strmblsd he/him]

im the cool uncle guys!!!! wanna go shooting???? or maybe go dirbike riding?????

[Through The Living Glass She/They]

18 minutes ago, Ookla the Invyzable said:

It’s okay, I’ve heard it’s a hard job.

Indeed it is.

 

2 minutes ago, Ookla The Loser OfThe Game said:

im the cool uncle guys!!!! wanna go shooting???? or maybe go dirbike riding?????

*gasp* I WANNA GO SHOOTING!!

[strmblsd he/him]

4 minutes ago, Through The Living Glass said:

Indeed it is.

 

*gasp* I WANNA GO SHOOTING!!

exactly I'm just the cool guy in the family..  all nonchalant (actually I  am the most challannt person you could ever meet)

[Immortal Platypus]

52 minutes ago, Ookla the Invyzable said:

*walks back in from playing volleyball*

I guess I’ve been replaced.

*sadly puts away wig*

it's ok Vyz. I'm here for you

[strmblsd he/him]

I  just started wheel of time I was reading and a character named wit was said and I legitimately gasped out loud in class then though oh my gosh I'm a idiot

[Through The Living Glass She/They]

4 minutes ago, Ookla The Loser OfThe Game said:

I  just started wheel of time I was reading and a character named wit was said and I legitimately gasped out loud in class then though oh my gosh I'm a idiot

LOL Strmblsd that's amazing

[strmblsd he/him]

2 minutes ago, Through The Living Glass said:

LOL Strmblsd that's amazing

it was actually so  funny

[Vyzkel He/Him]

25 minutes ago, Ookla the Arbiter said:

it's ok Vyz. I'm here for you

*wipes away tear*

Thanks

16 minutes ago, strmblsd said:

I  just started wheel of time I was reading and a character named wit was said and I legitimately gasped out loud in class then though oh my gosh I'm a idiot

I would’ve probably had the same reaction

[Immortal Platypus]

27 minutes ago, Ookla the Invyzable said:

*wipes away tear*

 

I gotchu.

44 minutes ago, strmblsd said:

I  just started wheel of time I was reading and a character named wit was said and I legitimately gasped out loud in class then though oh my gosh I'm a idiot

wait. there's a wit in WoT? Now I need to go reread

[Through the Living Shadow he/him]

48 minutes ago, Ookla the Arbiter said:

I gotchu.

wait. there's a wit in WoT? Now I need to go reread

Yea. I think he’s named Wit Congar, or smt like that. 

[Immortal Platypus]

9 minutes ago, Ookla the Dragonslayer said:

Yea. I think he’s named Wit Congar, or smt like that. 

oh yeah, him

[Through the Living Shadow he/him]

GUYS GUYS IF YA GET

The limit as a approaches infinity of

((((1 - x^a)/(1 - x))^0.5)/(x^(a/2-1)

THAT EQUALS

x/(x-1)^2

AND

THE ONLY NUMBERS THAT GIVE RATIONAL ANSWERS ARE SQUARES PLUS ONE

BUT

17 is 16+1

AND

170 is 169+1

IT’S SO WONDERFUL 

[Vyzkel He/Him]

30 minutes ago, Ookla the Dragonslayer said:

GUYS GUYS IF YA GET

The limit as a approaches infinity of

((((1 - x^a)/(1 - x))^0.5)/(x^(a/2-1)

THAT EQUALS

x/(x-1)^2

AND

THE ONLY NUMBERS THAT GIVE RATIONAL ANSWERS ARE SQUARES PLUS ONE

BUT

17 is 16+1

AND

170 is 169+1

IT’S SO WONDERFUL 

..what?

[Through the Living Shadow he/him]

Just now, Ookla the Invyzable said:

..what?

*STAB!*
MOOOOOOOOOOOOOOOOOONEY!

[Through The Living Glass She/They]

31 minutes ago, Ookla the Dragonslayer said:

GUYS GUYS IF YA GET

The limit as a approaches infinity of

((((1 - x^a)/(1 - x))^0.5)/(x^(a/2-1)

THAT EQUALS

x/(x-1)^2

AND

THE ONLY NUMBERS THAT GIVE RATIONAL ANSWERS ARE SQUARES PLUS ONE

BUT

17 is 16+1

AND

170 is 169+1

IT’S SO WONDERFUL 

Lol that's awesome

Just now, Ookla the Dragonslayer said:

*STAB!*
MOOOOOOOOOOOOOOOOOONEY!

Oh my goodness it's Mr. Krabs

[Vyzkel He/Him]

1 minute ago, Ookla the Dragonslayer said:

*STAB!*
MOOOOOOOOOOOOOOOOOONEY!

…oh

Can someone explain this to me?

[Through the Living Shadow he/him]

Just now, Ookla the Invyzable said:

…oh

Can someone explain this to me?

...there is no explanation. I just have a *twitch* habit of *twitch* stttaabbiiinnggg people.

(Unless u are talking about the math)

[Doomslug the Arcane she/her]

say what now

how did this go from nuggelanche to wrath stabbing people

[Through the Living Shadow he/him]

THIS TIME IT WAS SPIRITOFWRATH NOT ME

[Vyzkel He/Him]

16 minutes ago, Ookla the Dragonslayer said:

...there is no explanation. I just have a *twitch* habit of *twitch* stttaabbiiinnggg people.

(Unless u are talking about the math)

I’m talking about the math.

[Immortal Platypus]

1 hour ago, Ookla the Dragonslayer said:

GUYS GUYS IF YA GET

The limit as a approaches infinity of

((((1 - x^a)/(1 - x))^0.5)/(x^(a/2-1)

THAT EQUALS

x/(x-1)^2

AND

THE ONLY NUMBERS THAT GIVE RATIONAL ANSWERS ARE SQUARES PLUS ONE

BUT

17 is 16+1

AND

170 is 169+1

IT’S SO WONDERFUL 

i'm not gonna think hard enough to solve this

[Through the Living Shadow he/him]

OKOKOKOKOKOKOKOKOKOKOKOKOKOKOKOKOKOK

It doesn't help that I cant type math characters.

ORIGINAL TEXT (for reference):

The limit as a approaches infinity of

((((1 - x^a)/(1 - x))^0.5)/(x^(a/2-1)

THAT EQUALS

x/(x-1)^2

AND

THE ONLY NUMBERS THAT GIVE RATIONAL ANSWERS ARE SQUARES PLUS ONE

BUT

17 is 16+1

AND

170 is 169+1

IT’S SO WONDERFUL 

 

SO!

The first time I noticed a pattern from this was...

11!

When you take the square root of any number with only ones as its digits, it results in one of two things:

• In the case that there are an even number of digits, it is equal to the same number of threes, with the decimal in the middle. Then it has a bunch of numbers after that, but those aren't important.
• In the other case, with odd digits, it just gives you a bunch of numbers, stating with a one and a zero. (It does this because when you add a digit, you are essentially [there is a bit more to it) multiplying by the square root of ten.)

AND SO!

I wanted to make and equation to model this, because why not!

I was kind of stuck on how to represent the number of digits as the input, but then... I remembered sigma notation

(don't kill me, that's what it's called)

To convert number of digits to the actual number, I used

   

__a__

\

  >            (10^n)

/

______

n = 0

 

I didnt have the symbol, so it's weird.

That essentially just equals 10^0 + 10^1 + 10^2 + 10^3 + ... + 10^a, which converts it.

Now, this is a geometric series, which means I can represent the sum without sigma notation pretty easily.

So that there is equal to

(1 - 10^a)/(1 - 10)

or (1 - 10^a)/(-9), which I'll defined as f(a)

Now that I had that, I wanted to pursue the threes thing.

So if I take the square root of f(a),

it should give me that pattern that I saw up there.

And it did, but I wasn't really satisfied yet, as I wanted the first 3 to always be in the ones place, as... i dunno. huh.

So i divide it by 10 to the a/2 - 1 power, giving the effect I wanted.

 

AND I REALIZED IT CAN BE ABRA KEDABRA-ed BY A LIMIT!

So if you put int in a limit as a approaches infinity, you get 10/3, or 3.33333333333333333333333 repeating forever.

But I wasn't satisfied. I thought, well, a is no longer changing, so why not change the base of the exponent (currently 10, ya know)

So I got the original equation, which was kinda complex, and it was:

The limit as a approaches infinity, of

((((1 - 10^a)/(1 - 10))^0.5)/(10^(a/2-1).

Substituting x in for the 10s,

 

The limit as a approaches infinity of

((((1 - x^a)/(1 - x))^0.5)/(x^(a/2-1)

Next, I notice that the value of this at 5 was 2.5 exactly, or 5/2

COOL!

But I WAS NOT SATISFIED

I wanted to see why specifically 5 and 10 gave coherent answers, while all the other ones gave... incoherent answers. I found that 20, 100, and others didn't work, but then I stumbled upon 17, which did, giving 17/4

So the rational values I get have been:

5/2, 10/3, and 17/4

The numerator increases by a shifting value each time (more on that)

but the denominator increased by a CONSTANT VALUE! Success!

Still, I wanted to find a pattern for the numerator, and soon found that the amount it increased by increased by 2 each time, indicating a quadratic relation!

But this didnt mean that ut was quadratic, only that the numbers that were rational happened to be a shift on perfect squares. This, in turn, indicated that there was a square root!

5 - 1 = 4!

square root to 2, the denominator of 5!

10 - 1 = 9!

square root to 3, the denominator of 10!

And this repeated itself...

I HAD FOUND MY PATTERN!

That weird limit equation was equal to x / (x-1)^0.5

SUCCESS!

And to top it all off, both 17 and 170 worked in the equation. So there.

YES!