Proof that S_n = 2^n. By construction every number of a row goes into the next row twice, so the total sum doubles with each row and for the first row we have S^0 = 1.
Another way to think about it: the rows are nchoose0, nchoose1, ... ,nchoosen. This is finding how many subsets you have of an n-element set, which is 2^n.
Nice video! Speaking of Pascal's triangle, there's this amazing identity where you sum numbers diagonally and you get Fibonacci numbers, I think it'd make a great video too
In iran, we call this -the Khayyam Triangle- Khayyam,(an Iranian mathematician,) first discovered this, but later Pascal explained it further ;-)) Khayyam was truly a genius mathematician and we are proud of him.
It will be mistake to say Khayyam first discovered the Pascal triangle. Based on consecutive writings of Hindu scholars ranging from Pingala (2nd century BC) to Utpala (10th century) the basic binomial and recursive theme was well established in India. But Khayyam was a versatile talent - like Leonardo da Vinci - and I have read a few of his poems in english translation. Some of his poems, if they are written today by an Iranian or Arab poet would be called blasphemy. But I think his works as an astronomer show the real depths of his genius.
@@caesar_cipher Anyway, this is mathematics, the life of mathematics is as long as human life on earth. You can not say what mathematics really is and what it is not, because mathematics is the freest kind of human activity and can not be limited to anything. There are many people who have discovered beautiful seriese and patterns, but for some reason they have not been able to publish them. Mathematics has changed a lot throughout history and each time in a revolutionary way, Khayyam and Kharazmi are considered great mathematicians both in Iran and in world history. Yes, that's right, reading khayyam 's poems in Persian is very very enjoyable and informative 👍 I suggest you to read them :))
Ahahaha😂😂😂 5:50 pro gamer move 12:35 papa: "check out flammy 2" me: you got it😆 I mean, if people used your way of asking for something, i would be inclined on helping them in whatever way they want.☺ More often atleast.😆 (I was subscrided to Flammable maths 2 for some time now btw)
Ur math class In class the teacher says (a + b)^n = a^n + (nC1)a^n-1b + (nC2)a^n-2b^2 + … + (nCn-1)ab^n-1 + b^2, right? ;)I´m shure you can prove it you now algebra Me: going to sleep on the sofa
In the expansion of (x+y)^(n), for the ath term we have bx^(a)y^(n-a) (where b is the number of x^(a)y^(n-a) terms we get by simply expanding the product without simplifying). You have 'n' number of x's available and 'a' number of them can be picked to form the x^(a)y^(n-a) term (if x is not picked from one of the terms in the product y will be picked instead). So the number of x^(a)y^(n-a) terms is (nCa). Adding all of them we get (nCa)x^(a)y^(n-a).
Fun fact, if you go above, filling the hypothetical rows above the first one you get oscillating infinite series the first one is Σ(-1)^i which is unofficially considered 1/2, aka 2^-1. (I say unofficially because it's oscillating and not convergent, so it doesn't really have a value)
Now what happens to pascal's triangle when you go to 'negative' rows? If you assume 0s everywhere outside the positive row triangle, there's a single degree of freedom about each of the negative rows that you can fix however you wish, for instance by symmetry about the center line, or by setting all numbers on one side (either left or right) to continue being 0. Does the sum of 2^n continue to hold? Perhaps you need a special summation method like cesaro or abel?
Eh, that proof doesn't need induction, can be done very intuitively and simply, as a natural consequence of the algorithm of building the triangle. Starting with example: take row 4 [1 4 6 4 1]. Its values are calculated from row 3 [1 3 3 1] as (0+1), (1+3), (3+3), (3+1), (1+0), the sum being (0+1)+(1+3)+(3+3)+(3+1)+(1+0). Shift the brackets by one position. 0+(1+1)+(3+3)+(3+3)+(1+1)+0... Or in other words, 2*(1+3+3+1). Simply put, sum(row[n]) = 2*sum(row[n-1]) simply because every number from the previous row is added to the sum twice.
This was such a long discussion in Number Theory. So many secrets in binomial theorem and you have to use the summation form when proving some theorems.
Pascal triangle Best-of-seven playoff combinatorics baseball series 4-3 combination is most common 40 world series been played from total 109 total 37% of all world series, combinations of 4-0, 4-1, 4-2 is rare compare to 4-3.
My 10 year old sister had this problem where they were looking for the middle number in the 4th role and I guessed 6 but only cause I thought it was 11 in power of n, where n increases every row. Apperantly I was wrong. Ill have to explain this to her now.
Papa Papa Papa Papa Papa Papa Papa Papa Please do video series on Sets Papa Papa Papa Papa Papa Papa Papa Papa Maybe on 2nd channel? Papa Papa Papa Papa Papa Papa Papa Papa 🤓 Papa Papa Papa Papa Papa Papa Papa Papa
@@PapaFlammy69 hi i was viewing one of your video in which you asked to comment math exams with high failure rate so i desire you to try math part of JEE Advanced (an Indian exam) which is given a year after you pass JEE mains which is given by 12th graders. A good rank in JEE Advance is consider like 40%.
![[Discrete Mathematics] Binomial Theorem and Pascal's Triangle](http://i.ytimg.com/vi/stfqYC13bSY/mqdefault.jpg)
![[Discrete Mathematics] Binomial Theorem and Pascal's Triangle](/img/tr.png)
Proof that S_n = 2^n. By construction every number of a row goes into the next row twice, so the total sum doubles with each row and for the first row we have S^0 = 1.
For once a proof by induction is insightful because this exact reasoning can be found when using induction.
Another way to think about it: the rows are nchoose0, nchoose1, ... ,nchoosen. This is finding how many subsets you have of an n-element set, which is 2^n.
This lecture was like a hardcore math text alll results are left as exercises for the listener ;)
:D
:D:
I was just studying this yesterday - perfect timing. Thanks for the trivial exercise for the viewer too maximum simples
xD
Nice video! Speaking of Pascal's triangle, there's this amazing identity where you sum numbers diagonally and you get Fibonacci numbers, I think it'd make a great video too
yas!!
Papa-senpai noticed me again!
I'm sure you are probably aware but the Pascal holds a lot more secrets. Triangular numbers, powers of 11, etc.
In iran, we call this -the Khayyam Triangle-
Khayyam,(an Iranian mathematician,) first discovered this, but later Pascal explained it further ;-)) Khayyam was truly a genius mathematician and we are proud of him.
I'm starting to believe that this is true for every country, the italian equivalent is Tartaglia
Also a great poet
In turkey he is known for his poems about wine. He is a well known and loved figure here.
It will be mistake to say Khayyam first discovered the Pascal triangle. Based on consecutive writings of Hindu scholars ranging from Pingala (2nd century BC) to Utpala (10th century) the basic binomial and recursive theme was well established in India.
But Khayyam was a versatile talent - like Leonardo da Vinci - and I have read a few of his poems in english translation. Some of his poems, if they are written today by an Iranian or Arab poet would be called blasphemy. But I think his works as an astronomer show the real depths of his genius.
@@caesar_cipher
Anyway, this is mathematics, the life of mathematics is as long as human life on earth. You can not say what mathematics really is and what it is not, because mathematics is the freest kind of human activity and can not be limited to anything. There are many people who have discovered beautiful seriese and patterns, but for some reason they have not been able to publish them. Mathematics has changed a lot throughout history and each time in a revolutionary way, Khayyam and Kharazmi are considered great mathematicians both in Iran and in world history.
Yes, that's right, reading khayyam 's poems in Persian is very very enjoyable and informative 👍
I suggest you to read them :))
Great to see this really. Pascal's Triangle really helps with so many things, it's quite beautiful.
Ahahaha😂😂😂
5:50 pro gamer move
12:35 papa: "check out flammy 2"
me: you got it😆
I mean, if people used your way of asking for something, i would be inclined on helping them in whatever way they want.☺
More often atleast.😆
(I was subscrided to Flammable maths 2 for some time now btw)
This video is the preparation for the next one: fractional derivatives of imaginary rows of Pascal's triangle
xD
i´d love it
No joke I seriously needed this
nice :D
Ur math class
In class the teacher says
(a + b)^n = a^n + (nC1)a^n-1b + (nC2)a^n-2b^2 + … + (nCn-1)ab^n-1 + b^2, right? ;)I´m shure you can prove it you now algebra Me: going to sleep on the sofa
In the integers mod 2 the first binomial theorem is true
(a+b)^1=a^1+b^1
@@thedoublehelix5661 don't you mean mod n?
@@aidankwek8340 yeah, that´s induction :)
Some interesting math for you to look at( the silve rratio) ruclips.net/video/pqr7PLrJAzQ/видео.html
6:50 that escalated quickly
In the expansion of (x+y)^(n), for the ath term we have bx^(a)y^(n-a) (where b is the number of x^(a)y^(n-a) terms we get by simply expanding the product without simplifying). You have 'n' number of x's available and 'a' number of them can be picked to form the x^(a)y^(n-a) term (if x is not picked from one of the terms in the product y will be picked instead). So the number of x^(a)y^(n-a) terms is (nCa). Adding all of them we get (nCa)x^(a)y^(n-a).
Very cool Papa Flammy
thx! :3
Woah! I thought about this once. Thanks for uploading
:3 no problem!
Could u please make a video on ur masters thesis?
Now you can "prove" that the sum of all the rows in Pascal's triangle, going to infinity, is -1.
Is this a 3b1b reference
@@hoodedR yep.
@@alperakyuz9702 ayy nice... That's quite an old video isn't it
@@hoodedR No, it wasn't, actually. Look up 1 + 2 + 4 + 8 + ... in the Wikipedia.
Fun fact, if you go above, filling the hypothetical rows above the first one you get oscillating infinite series
the first one is Σ(-1)^i which is unofficially considered 1/2, aka 2^-1. (I say unofficially because it's oscillating and not convergent, so it doesn't really have a value)
Aight papa we need a video on your master's thesis
kk
Now what happens to pascal's triangle when you go to 'negative' rows? If you assume 0s everywhere outside the positive row triangle, there's a single degree of freedom about each of the negative rows that you can fix however you wish, for instance by symmetry about the center line, or by setting all numbers on one side (either left or right) to continue being 0. Does the sum of 2^n continue to hold? Perhaps you need a special summation method like cesaro or abel?
Eh, that proof doesn't need induction, can be done very intuitively and simply, as a natural consequence of the algorithm of building the triangle. Starting with example: take row 4 [1 4 6 4 1]. Its values are calculated from row 3 [1 3 3 1] as (0+1), (1+3), (3+3), (3+1), (1+0), the sum being (0+1)+(1+3)+(3+3)+(3+1)+(1+0). Shift the brackets by one position. 0+(1+1)+(3+3)+(3+3)+(1+1)+0... Or in other words, 2*(1+3+3+1). Simply put, sum(row[n]) = 2*sum(row[n-1]) simply because every number from the previous row is added to the sum twice.
2:25 RIP Him
As a computer science guy, seeing a math guy start an index at 0 makes me feel good in ways I prefer not to tell a priest about.
This was such a long discussion in Number Theory. So many secrets in binomial theorem and you have to use the summation form when proving some theorems.
Can anyone help me to solve the integral of sqrt(m - x + n/x) please?
Oh Boy! Old memories from kindergarten... Nice, old Pascal triangle!
:'D
Arrange Bell triangle, that's even more interesting.
pls make a vid showing us how you used it in your thesis plz
This is my favorite triangle
nice :3
Can somebody please remix the “in between, underneath” part XD
Beware of Fresh Toadwalker, he will declare it as the Gougu's Triangle.
Pascal triangle Best-of-seven playoff combinatorics baseball series 4-3 combination is most common 40 world series been played from total 109 total 37% of all world series, combinations of 4-0, 4-1, 4-2 is rare compare to 4-3.
Nice meme 0:09
The alternating version of the sum is also interesting, always equals 0
yas! :)
My 10 year old sister had this problem where they were looking for the middle number in the 4th role and I guessed 6 but only cause I thought it was 11 in power of n, where n increases every row. Apperantly I was wrong. Ill have to explain this to her now.
Just realised you have not covered Egorychev method
harstem + jerma
What happend with Instagram link?
whaddaya mean?
@@PapaFlammy69 It doesn't work for me.
weird, I'll look into it tomorrow!!
Wait where integral? No flame
Is
Are you playing DotA back then bruh? Imba was still used up until now in Dota2
4:43 papa why the fuck is not 0 in the naturals? My life is a lie :(
Fuck pascal triangle. All my homies use the pascal pyramid do find trinomial coefficients.
Tbh idk if this works, imma write some shit and check, cya
xD
Papa Papa Papa Papa Papa Papa Papa Papa Please do video series on Sets Papa Papa Papa Papa Papa Papa Papa Papa Maybe on 2nd channel? Papa Papa Papa Papa Papa Papa Papa Papa 🤓 Papa Papa Papa Papa Papa Papa Papa Papa
The meme is something else :/
hi
hey :v
u cute; ❤️🤸
do you like physics?
kinda
@@PapaFlammy69 hi i was viewing one of your video in which you asked to comment math exams with high failure rate so i desire you to try math part of JEE Advanced (an Indian exam) which is given a year after you pass JEE mains which is given by 12th graders. A good rank in JEE Advance is consider like 40%.