>>4922049

I want to keep posting on SEQUENCES, not to distract but rather to keep us thinking new thoughts- kind of brain snacks to keep us thinking creatively and maybe see things in a new way, and HOPEFULLY find a sewuence or a key. (brainfood below)

A RULE:

A Sequence usually has a Rule, which is a way to find the value of each term.

Example: the sequence {3, 5, 7, 9, …} starts at 3 and jumps 2 every time:

{3, 5, 7, 9, …}

As a Formula

Saying "starts at 3 and jumps 2 every time" is fine, but it doesn't help us calculate the:

10th term,

100th term, or

nth term, where n could be any term number we want.

So, we want a formula with "n" in it (where n is any term number).

So, What Can A Rule For {3, 5, 7, 9, …} Be?

Firstly, we can see the sequence goes up 2 every time, so we can guess that a Rule is something like "2 times n" (where "n" is the term number). Let's test it out:

Test Rule: 2n

n Term Test Rule

1 3 2n = 2×1 = 2

2 5 2n = 2×2 = 4

3 7 2n = 2×3 = 6

That nearly worked … but it is too low by 1 every time, so let us try changing it to:

Test Rule: 2n+1

n Term Test Rule

1 3 2n+1 = 2×1 + 1 = 3

2 5 2n+1 = 2×2 + 1 = 5

3 7 2n+1 = 2×3 + 1 = 7

That Works!

So instead of saying "starts at 3 and jumps 2 every time" we write this:

2n+1

Now we can calculate, for example, the 100th term:

2 × 100 + 1 = 201

Brainfood for SEQUENCES:

MANY RULES:

But mathematics is so powerful we can find more than one Rule that works for any sequence.

Example: the sequence {3, 5, 7, 9, …}

We have just shown a Rule for {3, 5, 7, 9, …} is: 2n+1

And so we get: {3, 5, 7, 9, 11, 13, …}

But can we find another rule?

How about "odd numbers without a 1 in them":

And we get: {3, 5, 7, 9, 23, 25, …}

A completely different sequence!

And we could find more rules that match {3, 5, 7, 9, …}. Really we could.

So it is best to say "A Rule" rather then "The Rule" (unless we know it is the right Rule).

NOTATION USED:

To make it easier to use rules, we often use this special style:

sequence term

xn is the term

n is the term number

Example: to mention the "5th term" we write: x5

So a rule for {3, 5, 7, 9, …} can be written as an equation like this:

xn = 2n+1

And to calculate the 10th term we can write:

x10 = 2n+1 = 2×10+1 = 21

I like the one below because it involves differences between numbers(maybe we can use deltas?)

ARITHMETIC SEQUENCES:

In an Arithmetic Sequence the difference between one term and the next is a constant.

In other words, we just add some value each time … on to infinity.

Example:

1, 4, 7, 10, 13, 16, 19, 22, 25, …

This sequence has a difference of 3 between each number.

Its Rule is xn = 3n-2

In General we can write an arithmetic sequence like this:

{a, a+d, a+2d, a+3d, … }

where:

a is the first term, and

d is the difference between the terms (called the "common difference")

And we can make the rule:

xn = a + d(n-1)

(We use "n-1" because d is not used in the 1st term).

GEOMETRIC SEQUENCES:

In a Geometric Sequence each term is found by multiplying the previous term by a constant.

Example:

2, 4, 8, 16, 32, 64, 128, 256, …

This sequence has a factor of 2 between each number.

Its Rule is xn = 2n

In General we can write a geometric sequence like this:

{a, ar, ar2, ar3, … }

where:

a is the first term, and

r is the factor between the terms (called the "common ratio")

Note: r should not be 0.

When r=0, we get the sequence {a,0,0,…} which is not geometric

And the rule is:

xn = ar(n-1)

(We use "n-1" because ar0 is the 1st term)

Triangular Numbers

1, 3, 6, 10, 15, 21, 28, 36, 45, …

The Triangular Number Sequence is generated from a pattern of dots which form a triangle. starting with one dot by itself(gonna have to imagine a tringle around it), then 3 dots which form a triangle, then a triangle with three dots on ea. side, then a triangle with four dots on each side with one in the middle, then a triangle with five dots on each side with three dots in the shape of a triangle in the middle, and so forth.

By adding another row of dots and counting all the dots we can find the next number of the sequence.

But it is easier to use this Rule:

xn = n(n+1)/2

Example:

the 5th Triangular Number is x5 = 5(5+1)/2 = 15,

and the sixth is x6 = 6(6+1)/2 = 21

SQUARE NUMBERS SEQUENCE:

1, 4, 9, 16, 25, 36, 49, 64, 81, …

The next number is made by squaring WHERE IT IS in the pattern!

This needs some explanation but it is genius! Think POSiTION of the number! First position =1, second position =2, third position =3, etc

1 squared=1

2 squared=4

3 squared=9

4 squared=16

Rule is xn = n2

I love this sequence code. It is sooo Q. Now how is that for a sequence?

>>4948748

That is an exciting theory Anon. How does MIL find ememy's position to call in an airstrike against them?

By TRIANGULATION.

You need 2 different observers in different locations (with known coordinates). Each calls in an azimuth reading on a compass toward the target from their respective locations).

Field Artillery draws a line along the azimuths given from the two observors" known locations on the map and where the 2 azimuths cross- they get the location(coordinates) of the target on a map.

CUBE NUMBERS SEQUECE:

1, 8, 27, 64, 125, 216, 343, 512, 729, …

The next number is made by cubing WHERE IT IS in the pattern.

Rule is xn = n3

This is another exciting sequence because it involves LOCATION of the number.

1st location =1

2nd location =2

3rd location =3

etc.

1 cubed = 1

2 cubed = 8

3 cubed = 27

etc.

FIBONACCI SEQUENCE:

This is the Fibonacci Sequence

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …

The next number is found by adding the two numbers before it together:

The 2 is found by adding the two numbers before it (1+1)

The 21 is found by adding the two numbers before it (8+13)

etc…

Rule is xn = xn-1 + xn-2

That rule is interesting because it depends on the values of the previous two terms.

Rules like that are called recursive formulas.

The Fibonacci Sequence is numbered from 0 onwards like this:

n = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 …

xn = 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 …

Example: term "6" is calculated like this:

x6 = x6-1 + x6-2 = x5 + x4 = 5 + 3 = 8

>>4949532

Sorry, we anons can live compartmented from eachother. What is the ATS thread?

>>4949886

Sure would be nice to have some GovtAnon who knows codes stop by to ease us in the right direction.

I feel good about this group here.

I don't know Excel but am hopeful some Anon here will make a spreadsheet that makes a sequence stand out. It's time to solve the map. For my money, this is the thread to watch.

>>4950168

>Q/POTUS-1

>Q/POTUS-2

>Q/POTUS-3

>Q/POTUS-4 [10]

>Q/POTUS-5

I must admit my autism is not with deltas and time stamps BUT- what is the reference of these Q/POTUS deltas? When?

Are they literally the first 5 Q/POTUS deltas? Even if they are not, amd maybe they are from a new point, then If the 4th is known, then go to the next, (5th?) and give us the delta. For the other three, try going back from the 4th. Kind of like a crossword puzzle- always work from what is known. Fill it in from there.

>>4948748

I like the idea of coordinates on a map, especially the first 2.

So much to ponder. We are supposed to find the legend/key on the map. And I believe we had it as early as Post#128 in the graphic.

There IS another picture provided in the graphic.

It is Washington Crossing the Delaware.

Interestingly, I found there is a Washington Crossing National Park with a website. I hesitated to post this here because I didn't want to slide this thread, but there is a map on the website. (pic included here).

On the map are 2 highways that are marked by KEYSTONE symbols. Highway 32 runs East-West. Highway 532 runs North-South.

At this point I feel we need to share anything we can, if it might be of significance.

See for yourself.

>>4950548

That sounds great. Q Cartographers thread.

And I do have another map of that area covered with them.

>>4952619

Thank you Anon. Will carry on.

>>4952774

Anon, am just posting this, as before, not to try to steer away from any good ideas, but to look at different things for inspiration.

My first real exposure to this map was in Paris. The maps are genius in that they show connections of various routes(topics?) but show connections and are color-coded(as map keys are).

If you want to go to a stop at the end of the line, from across the city, you must make "connections" at hubs or line divisions. In the subway stations you are guided by color-coded signs with numbers to get to the proper boarding deck- on that line which can go in two directions.

Could help organize.

>>4952612

I start at Post #128. It is apparent that, as of that post, Q tells us that it is possible to look for a legend/key. In the middle of questions about maps, Q asks, what is a series.

Of interest are the two lines which state:

Future provides Past.

Map provides Picture.

(This is not how Q normally posts

Future proves past.)

Notice the extra letters.

Future provIDes Past.

Map provIDes Picture

Some of us think the Sequence may be a "numerical" key/legend to read the map. This can possibly mean "reading" deltas or time stamps with more meaning.

Also looking into the Q clues re "picture" as a possible Legend.

ie Map provides picture.

Paint the picture.

Big picture

In Post #128 there are a few.

One of them was painted.

Others taken at 40,000ft?

This thread is where ExcelFags may chart some groups of time stamps and deltas that may, if we are lucky, hint at a sequence.

That is why I posted the Sequence theory.

The SET theories may also show that a subset of numbers that is contained in a larger set of sequences, might single it out as a common sequence.

In some cases, if you remove a subset, the larger set cannot exist.

Picture a Venn diagram with a large circle containing different numbers as the SET. Then picture a smaller circle within that set, a SUBSET which contains some of the SET. If you remove the subset, the larger set that once contained those numbers cannot exist. This is a type of KEYSTONE.

Some of us think the keystone may be numerical.

Have searched all kinds of sites on maps, how to read them, basic components of a map. None that I have searched has a "keystone" aid to reading map.

Yes, we have found some maps that have some highways on a map of the Washington Crossing Historic Park designated by a keystone symbol, but are not sure if it's a rabbit hole.

But leave no stone unturned I say.

Ask yourself this, Why did Q ask, What is a sequence? Why is this relevant? (in the middle of map-related clues)? I refer to Post#128.

>>4953250

>kek anon, what made you chose that map?

>curious

I happened to be following a hunch about the "Map provides picture". I decided to look up Washington Crossing the Delaware because it was provided in the Post#128 map. I found a website for the Washington Crossing Historic Park. I may have come up in listing maps of the area around Trenton.

I had also searched the route taken by Washington from the crossing point to Trenton. Interesting thing, Pennsylvania marks some highways with keystone symbols. Funny thing.

>>4950239

I MIGHT see a predictive sequence there:

You have sequence of:

15; 5; 3; 10; 5

If it is a PARTIAL sequence, (like on IQ tests) one could complete the sequence, in full as:

15; 5; 3; 10; 5; 2; 5; 5; 1; 0; 5; 0

With a disclaimer, this is just tooling around, here is how, breaking them into subsets.

15/5=3

10/5=2

5/5 =1

0/5 =0

—————–

If I was taking an SAT or IQ test, this is how I would have completed the series.

>>4956446

So, my completion of the first 5 Q/POTUS deltas solved by Anons would be:

Q/POTUS-6 [2]

Q/POTUS-7 [5]

Q/POTUS-8 [5]

Q/POTUS-9 [1]

Q/POTUS-10 [0]

Q/POTUS-11 [5]

Q/POTUS-12 [0]

Remember, there can be more than one solution.

My rule here is: take the first number divided by the second number to get the third. Repeat with the next numbers.

Let's see what happens.