Why don't they teach Newton's calculus of 'What comes next?'

[Music] welcome to another mythology video let's get going today with a very familiar mathematical guessing game [Music] you all know what i'm about to say right yep you guessed it what's next pretty crazy isn't it definitely not the kind of sequence you want in the middle of an iq test of course just looking at the first five terms 1 2 4 8 16 most people would bet their life that what we're dealing with here are just the good old powers of two but then very surprisingly things break down badly and the numbers start looking quite random despite all this i promise you cross my heart and hope to die i promise you that what's up there is a sequence that arises naturally in mathematics very curious but if the sequence arises naturally how can we discover the key to the pattern well next time you're confronted with a mystery sequence you could try doing this two minus one equals one four minus two is two eight minus four is four sixty minus eight is eight thirty-one minus sixteen is fifteen and so on but why stop there two minus one is one four minus two is two and so on that's it of course not two minus one is one and so on and one more time very interesting we end up with a row of ones and so there appears to be a hidden pattern after all now assuming that these ones continue we can actually calculate what the next number of our mystery sequence is how well 15 minus 8 is 7 is the same as saying that 8 plus 7 is 15. green plus orange is purple and of course that is true for any triangle in our grid there eleven plus five is sixteen five plus one is six but that means that if the ones in the bottom row continue and the purple must be six plus one is equal to seven twenty-two plus seven is twenty-nine sixty-four plus twenty-nine is ninety-three and there's the next number of our original sequence 163 plus ninety-three that is 256. interesting another power of 2. is that a coincidence well let's see chucking another 1 in the bottom row and calculating upwards we find that the next number in our mystery sequence is definitely not a power of two so it's still not clear yet what's going on with our mystery sequence but assuming the bottom row of once is the key and goes on forever we can still continue and reconstruct all infinitely many terms of the mystery sequence very cool and a challenge for the hardcore ones among you can you find another power of two in this sequence and is there a pattern where the powers of two occur record your findings in the comments okay we've discovered a cool trick for trying to decipher what's next iq puzzles but of course we're not done yet what we would really like is a simple rule a formula for our mystery sequence that would be great right and is it actually possible to somehow figure out all the formulas for all the sequences that end in constant rows that would be even more amazing right now a super cool way to organize our ideas into a systematic attack on these questions begins with the realization that what we're actually doing here is calculus wait what calculus all that dydx stuff yep pretty much the calculus of sequences which is what we're about to explore is a parallel universe to the familiar calculus taught in school in this strange new calculus sequences play the role of functions differences play the role of derivatives sums play the role of integrals difference equations mirror differential equations and so on and this may surprise many of you real world calculus problems are often solved using this strange sequence calculus and all of your familiar school calculus follows quite easily from the calculus of sequences super cool stuff okay today's mission to give you a crash course in the calculus of sequences and its many nice applications to begin i'll demystify our what's next puzzle solver and then we'll look at the amazing gregory newton interpolation formula sounds grand and it really is plus there will be amazing alternate reality views of many of our all-time favorites such as pascal's triangle fibonacci sequence maclaurin series and so on as usual lots and lots of amazing mats to look forward to if you don't know any calculus i'd say just buckle your mathematical seat belts follow along and don't worry about the calculus references too much i'll keep it as self-contained for you as possible all good ready great and let's go [Music] okay screw calculus is about functions so where are the functions in this setup well let's call the original sequence f for function then we can think of the terms of the sequence as the values of the function at 1 2 3 4 5 etc makes sense right first term second term third term and so on well actually instead of starting with one two three and so on in this part of the mathematical world we start at zero with a zeros term mathematicians don't you just love them anyway with that hiccup sorted we can graph our sequence just as usual there f 0 that was 1 f 1 f 2 and so on ok what about the second row well the first term of the second sequence is the difference f one minus f of zero and so on and in the diagram this difference corresponds to this vertical distance but there's a second way in which this difference makes an appearance in this diagram and the second way is what's important for our story the difference is also the slope of this line segment right the slope of this blue segment is orange rise over a run of 1 and so is equal to the orange difference up there fantastic the difference in the second row are just the slopes of these segments so definitely very calculating right remember in school calculus the derivative of a function at a point gives you the slope of the function at this point so in our sequence world the second row sequence is really the derivative of the first row but not surprisingly the official name for this sequence derivative is difference so the second row is the difference of the first row and is abbreviated by a greek capital delta a greek d making all my greek viewers super happy if we're dealing with several roles we put a little superscript to distinguish between them there the first difference the second difference and the third difference of the original sequence later we also go upwards as well creating undifferences or anti-differences of course that is the counterpart to enter differentiation standard integral calculus definitely something to look forward to now let's just quickly single out the first entries in each row those entries there say you know nothing about a different scheme except for those entries then you can reconstruct part of the different scheme as we did before let me quickly animate the reconstruction process for you the green plus orange is purple as before and then it keeps going and as you've already seen if you also know all the entries of the last row then you can reconstruct the whole different scheme like this right so for example for our original mystery sequence these numbers in front and the fact that one at the bottom repeats to the right is enough to reconstruct the whole difference scheme which of course includes the mystery sequence itself at the top remember that one for later or good so far pretty easy right okay to get a feel for how all this works let's calculate the formulas for the difference of some simple but super important sequences [Music] remember our mystery sequence starts with five powers of two let's have a look at the part of the different scheme that comes from differencing those powers of two what about the second row do you notice something they had a powers of two again so it looks like the difference of the powers of two is again the powers of two let's check that this is really true okay so the formula for the difference of our sequence is this just a difference of two consecutive terms of the sequence right now we're talking about the powers of two so in this case f of n is simply equal to two to the power of n then the difference created from two to the power of n is this [Music] so 2 to 12 n is its own difference very nice does this ring a bell yep this is very reminiscent of the exponential function e to the power of x being its own derivative in standard calculus and indeed 2 to the power of n will end up playing the role of the exponential function in our strange new calculus ok what other sequences could we try well in standard calculus we get going with x squared so let's try squares in this case we put f of n is equal to n squared and the sequence of squares is this 0 1 4 9 16. now the difference of the squares is this aha it's just the odd numbers very cool the difference of the squares are the odd numbers let's also do the algebra on autopilot of course two n plus one right two times whatever is even plus one is odd numbers cool and two n plus one that's almost the same as with standard calculus right there the derivative of x squared is 2x let's keep on going what's the second difference of the squares well pretty obvious three minus one is two five minus three is two and so on all twos okay so the second difference of the squares is just two and of course exactly the same is true in school calculus the second derivative of x squared is two so all in all very similar but not quite the same which is a bit annoying but that turns out to be a surprisingly easy fix which we'll get to soon anyway notice that this constant row at the bottom of the different scheme for n squared is of course also what was special in the case of our mystery sequence in the case of our mystery sequence the fourth difference is equal to one if you're familiar with calculus you can probably make his guess now what sort of functions spawned our mystery sequence got it yes no well let's keep on going okay so the second difference of the squares is constant what should we try next yep just like in school how about cubes let's see whoa nice fizzles out too this time the third difference is all sixes one more powers of four cool well and we didn't expect anything else at this point the force difference is also constant of course bells will also be ringing for everybody who recently watched my video on mercenaries miracle if you haven't watch it after this video also those constant in the bottom rows are very familiar aren't they so for example 24 is the fourth difference of n to the power of four and similarly we get this okay one two six twenty four well i'd say pretty much everybody here would again be prepared to bet their life that these are just factorials and no i'm not trying to trick you into gambling away your life that's actually correct and easy to prove we really do get the factorials forever and ever after and we'll see these factorials showing up all over the place shortly very nice stuff isn't it here are some more easy but also very important insights start with a sequence say n cubed together with its different scheme what if we had started here is double the sequence with two times n cubed well for example the difference of these two terms two times eight minus two times one that's just two times eight minus one which is just two times the original difference 7 and of course the same is true for all other differences so it's clear that when you multiply the terms of a starting sequence by some constant then the new difference scheme will just be that constant times the original difference scheme easy right here's another interesting important rule which is also very easy to prove what if you add two sequences together term by term to create a new sequence well then the difference scheme of this new sequence is just the sum of the two starting difference schemes so just superimpose and add up like this nice nice nice nice nice then just like in school calculus we can use these two simple rules to build a different scheme for any polynomial sequence for example to find a different scheme for this polynomial sequence here we simply multiply the different schemes of the powers n cubed n squared n to the power of 1 and n to the power of 0 which is just 1 by those coefficients 8 2 3 and 4 and add up okay now to get the difference scheme for our polynomial sequence we just have to superimpose and add up this means that the polynomial difference scheme we're after has the same number of rows as that of our highest power the last row is exactly the same constant sequence as that of our largest power just multiplied by 8. very interesting isn't it and this means that the difference schemes of absolutely all polynomial sequences end in constant rows how cool is that in fact it turns out that it is exactly the polynomial sequences that eventually result in constant rules this tells us for example that our original mystery sequence must be polynomial also its difference scheme has one more row than the scheme for our degree three polynomial up there so it's reasonable to suspect that the polynomial of our mystery sequence is of degree four but how do we find this mystery polynomial in our mystery sequence well once you know or suspect that the rule is polynomial there are a couple of different ways one natural but tedious one which you may know from school is to solve a system of linear equations then there's also something magical called the lagrange interpolation formula which builds the polynomial in one easy step i also have to talk about that one at some point today i want to show you a third way to mathematical nirvana this third way has the grand name of gregory newton interpolation formula or just gregory nugent formula or just newton formula somehow the big shots always get the credit right so we'll stick up for the little guy and refer to it as the gregory newton formula get ready for an avalanche of aha moments and beautiful mathematics [Music] okay sleeves rolled up and down to work over there is the start of another different scheme that ends in a constant row now let me show you the gregory newton formula in action the super simple procedure for finding a polynomial rule that fits the sequence at the top ready to be amazed here we go [Music] and that's all there is to it does it work newton gregory aren't people to argue with but let's check anyway plug in zero and the formula spits out one plug in one and we get eight plug in two it spits out 17 and similarly for n is equal to three and n is equal to four brilliant okay all this was quite fast so let's spool back a couple of steps the spool spool it's an amazingly simple formula you just have to multiply the highlighted numbers at the start of each row by those bracket things the binomial coefficients most of you will probably remember from school that those binomial coefficients can be written like this at this point you then just have to expand and simplify to arrive at the compact formula okay back again to the original formula before we do anything else let's complete our original mission and use the gregory newton formula to write down the polynomial rule for our mystery sequence for mystery sequence all the highlighted coefficients are ones and so the formula we're after is this okay super pretty mission accomplished but of course this is mythology we prove things here i have to prove to you that the gregorian newton formula really works and i'll do that in the last part of the video but before that i want to mine the gregory newton formula for some mathematical beauty and great applications i'll now show you five insights into our new sequence calculus provided by the gregory newton formula [Music] first you can use the grand newton formula to solve any what's next riddle for example let's say we have a sequence that begins with the mathematical super constants pied and find an e so a really really mean iq test question would be what comes next easy we can make up the difference scheme and plug it into the gregory newton formula the resulting polynomial will spit out pied and phi and e and then whatever spits out next that's our answer [Music] there plug in zero you get pi plugin one you get five plugin two you get e so what comes next well plug in three and there's your answer first python 5 and e and then next is 3 minus 3 5 plus pi really are you willing to bet your life on this answer is that really what comes next well yes and no gregory newton has given us a simple formula and that formula spat out 3 minus 3 5 plus pi but you could similarly argue for any number to come next right so what's your favorite number today minus 666 for mischievous reasons and so what if we want 666 to come next not a problem we simply go through the gregory newton procedure starting with these four terms pi phi e and 666. okay okay not pretty but it works this example shows you that the whole what's next game is fundamentally really silly if anything can be an answer then of course nothing is an answer so on your next iq test you have my permission to answer whatever you like comes next let me know what happens on the other hand in the proper scientific context what's next is a really important question right perform a couple of experiments to explore naturally occurring phenomenon and then try to guess a natural and general rule based on the outcomes but of course you don't stop there you then also have to somehow justify that your guess is correct and so far that justification is also missing from the discussion of our original mystery sequence as i said the sequence arises naturally in mathematics so to make sure that our formula is correct we still have to prove that the formula correctly captures the mathematical context so i guess it's time to give you that context here it is take a circle and put some dots on the perimeter there one two three four five six dots draw all the lines connecting all pairs of dots then these lines will cut the circle into regions with the number of regions depending upon the location of the dots then with n dots the maximum possible number of regions is the nth term of our mystery sequence there 1 2 4 8 16 and 31 regions surprisingly have we found the formula for that region counting yes as it happens we have but what we haven't done is prove that our formula works forever and ever after the pretty proof is a bit too much of a tangent for this video and so i'll just link to it in the description make sure to check it out it will be on the exam okay [Music] now take any sequence of numbers whatsoever gregor and newton will always produce a formula for this sequence just usually the formula will have infinitely many terms let me explain using a super important and super pretty example remember 2 to the power of n the sequence of the powers of 2 is its own difference this means its difference scheme has infinitely many rows and all these rows will be identical there that's a different scheme for 2 to the power of n all rows are identical 1 2 4 8 16 etc and there are infinitely many of these rows and all of these infinitely many rows start with a 1 and that means that the gray newton formula for 2 to the power of n is just the sum corresponding to these infinitely many highlighted ones so the sum of the infinitely many binomial coefficients so plugging in 0 1 2 3 4 and so on this infinite sum is supposed to spit out the powers of two really doesn't seem likely does it okay let's check let's cross our fingers and expand those infinitely many binomial coefficients let's see what happens when we plug in n is equal to 2. look at that 2-2 that's 0 and of course you get those 2-2s in all the higher order terms which means that all these terms are zero so this infinite sum is really just the finite sum in disguise works and the same is true in general let's have another look at our example just like for this specific example the gregor newton series of any infinite sequence has infinitely many terms however when you evaluate this infinite formula at a positive integer all but finitely many terms of the formula will be equal to zero pretty magical and so using such a scary looking infinite formula is actually not a big deal fantastic but there is more pretty stuff here much more for example pondering this formula for 2 to the power of n for a moment some of you will be reminded of another really iconic maths formula remember we said that 2 to the power of x is the counterpart of e to the power of x in school calculus well the mega famous maclaurin series for e to the power of x goes like this have a look at it pretty much exactly the same except for those n times n minus 1 times n minus 2 etc products on top right but now since 2 to power of n is the counterpart of e to the power of x this suggests that our sequence calculus those n times n minus 1 times n minus 2 products are somehow the counterparts of the nth powers in school calculus intriguing let's have a closer look these special products are called falling powers for example n times n minus 1 is called n to the falling 2 and is abbreviated like this so it's indicated just like n squared but with the 2 underlined and similarly with the other falling powers that n here is also n to the falling 1 and the 1 in front is also end to the falling zero very nice also the name falling powers cooler and even nicer those falling powers really behave just like the normal powers as far as differencing is concerned remember the derivative of x to the power of m equals now mantra m times x to the power of m minus 1. the same is true for the falling powers the difference of x to the falling m is m times x to the falling m minus one cool a little challenge for the keen among you prove this differencing formula for the falling powers leave your proofs in the comments not terribly difficult if you have proofs for breakfast also in general the fix for the annoying factor differences of normal powers don't behave quite as nicely as their derivatives is to express things as much as possible in terms of these falling powers in sequence calculus anyway there's more as i said our falling powers formula for 2 to the power of n works for n is equal to 0 1 2 3 any non-negative integer but much more is true it turns out that we can plug in any positive number any positive real number whatsoever and the formula still works that's amazing isn't it for example plug in x equal to one half there unlike when we plug in 0 1 2 3 etc none of the terms in this infinite series is 0. this is a real infinite sum but still if you add up all those infinitely many terms you really get to the power of one half and of course to the power of one half is root two and so among other things taking finite chunks of this formula gives us the way to approximate root two but a word of caution of course there are tons of real valued functions that will agree with 2 to the power of xl 0 1 2 3 and so on we just join the dots any which way and obviously for any such jointed dots function gregory newton will produce the exact same formula as for 2 to the power of n but only for 2 to the power of x will this formula work for all positive real numbers very interesting isn't it what makes 2 to the power of x super special in this respect what happens for negative x for example see what you get for x equal to minus 1 and x is equal to -2 for a really pleasant surprise especially if you watched one of the one plus two plus three videos now in general it's very tricky to determine what one of these gregory newton series does when x is not one of the non-negative integers some real mathematical monster mystery lurking in this area that are well beyond the scope of this video [Music] okay i also promised you that we'll be exploring what it means to anti-differentiate or integrate in the sequence calculus to motivate this let's have a look at the different scheme for the squares again well since differencing the counterpart of differentiating produces rows going down the counterpart of anti-differentiating should produce rows going up okay so how can we extend our difference scheme going up what numbers should go in those spaces here well let's just make an arbitrary choice for the first space on the left let's make this zero then the rest is forcing us by our usual reconstruction procedure zero plus zero zero plus zero plus one zero plus zero plus one plus four zero plus zero plus one plus four plus nine and so on uh-huh so while the opposite of the derivative is the anti-derivative the opposite of the difference is the anti-difference in other words the sum where summing the elements of our sequence makes sense and this means that we can also use the gregory newton formula to find formulas for the sums of sequences that's also one of the main uses for this type of calculus finding nice formulas for crazy sums of sequences here's an easy example let's calculate the formula for the sum of the squares in front of us highlight the leading entries yeah and then gregory newton tells us to just multiply these highlighted numbers by the binomial coefficients and add so plug in zero you get zero plug in one you get zero plus zero that's zero again plug in two you get one well those two ones at the beginning of our sum sequence aren't doing much are they so instead of using the highlighted zero zero one two to build the formula it's better to shift one along and begin with these guys to build a formula zero one three and two now go again plug in zero you get zero 0 0 plug in 1 you get 1 and so on much better then expanding and simplifying the sum gives you the familiar sum of squares formula many of you will have seen that one before i think beautiful and easy as i said in generally gregory newton is an incredibly useful tool for finding formulas for crazy complicated sums if you like to get an idea of what's possible i give some links in the description so you can wander down this particular rabbit hole and one more easy remark remember we started something by making our choice we chose the first entry to be 0. how would this change if we had chosen another number let's say c easy just replace the leading zero and everything that follows by c and that amounts to well just adding c to all entries of the first row and this says that our sequence sum just like the anti-derivative in school calculus is uniquely determined up to a plus c how many points have we lost in school by forgetting to write plus c when answering an integration problem well i lost quite a few anyway very neat everything's falling into place that's the end of it actually no there's one more important insight i have to share with you at this point remember this picture from earlier on remember that those differences also have a geometric meaning that mirrors the geometric meaning of the derivative in school calculus those differences are the slopes of the blue segments what's the geometric meaning of our sums there those sums at the top i'm sure that if you know some calculus you can guess the answer you've got it well f of 0 f of 1 f of 2 those are just those vertical distances but they are also those rectangle areas right those rectangles all have a base length equal to one and so their areas are equal to their heights but that means that our sums are just those staircase areas under the blue curve but now remember that in school the main use of antiderivatives is to find the area under functions aha those sums are counterparts of definite integrals in school calculus yet another puzzle piece falling into place very satisfying don't you think are you starting to feel a little bit like alice in wonderland [Music] those of you who've done some calculus in school will remember that differentiating is easy and integrating is hard what is harder than integrating calculus well solving differential equations let me also quickly show you what that looks like in sequence calculus to do that let's have a look at the different scheme of everybody's favorite sequence the fibonacci sequence well everybody's favorite except for marty my friend marty he hates the thing you'll have to ask him okay here's the start of the fibonacci sequence remember the fibonacci sequence starts with zero and one and then every term is the sum of the two preceding terms zero plus one is one one plus one is two one plus two is 3 and so on okay here are the first and second differences of the fibonacci sequence have a close look can you see something remarkable happening yup the whole sequence replicates just shift it over by one there 0 1 1 2 3 again and again very nice very similar to what happens with 2 to the power of n right have a look at the three entries along this diagonal line here these are just three consecutive terms of the fibonacci sequence well with the shifting in place that makes sense and so two plus three is equal to five three plus five is equal to eight and in general the second difference plus the first difference is the sequence itself or reshuffled a bit we get this equation here very nice what we've isolated here is a difference equation the fibonacci sequence is the solution to this difference equation together with the initial values 0 and 1. the niftiest part is that you can also solve this difference equation pretty much exactly as the corresponding now warning warning big words coming up but don't worry about them you can solve this difference equation pretty much exactly like a linear second order homogeneous differential equation and when you do this you actually end up with benay's mega famous formula for the fibonacci numbers pretty spectacular formula all my disciples who did calculus with me here at monash uni should be able to do this can you for those of you interested in the details i linked to a write-up in the description of this video just to give you a taste for what's happening remember that the fibonacci sequence gets repeated in its own difference which is very similar to what's happening with 2 to above n being its own difference this suggests the general formula for the fibonacci sequence may also be a combination of exponential functions which ones well there's a standard way to play that guessing game exactly mirroring what is done in differential equations it all works out easy coming down to the roots of the appropriate quadratic equation following this lead gets us binai's formula right binay's formula is basically the difference of two exponential functions again very pretty stuff and another important example of how our sequence calculus can be used to find simple formulas for sequences [Music] so pretty much everything in school calculus has a counterpart in our sequence calculus here just a couple more highlights there on top is the product formula for differentiation and below is the product rule for sequence calculus prettier and there the crown jewel the fundamental theorem of calculus and below it the fundamental theorem of sequence calculus challenge for you prove this fundamental theorem of sequence calculus give it a try real fun especially if you like telescoping sums not that hard actually and this one well i already discussed that one on top of the maclaurin formula and below it the gregory newton formula in all its formulaic glory featuring the falling powers i have to make a new t-shirt with that formula here before i get into the proof of the gregory newton formula let me expand a little bit on my claim that all the sequence calculus is one way you actually apply school calculus in real life numerically let's say that's a complicated function arising from an experiment that you want to torture with calculus more often than not we will be given this function in a digitized way so we're given an approximation of the function in terms of the red points now to be able to apply our sequence calculus we simply have to scale this picture so that the small step size becomes one then apply sequence calculus and then account for the scaling in the result for example let's say we want to calculate an approximation of this area scale up and a line note down the values of the function in front of us the values at the red dots do the sequence integral then as we've seen this entry here will give this green area and of course this green area is a rough approximation of the area that we're interested in at this step and then we scale back to get the area that we are really really interested in okay nice but maybe not that super impressive but there are a couple of ways in which we can do a lot better by computing the differences now we apply gregory newton and if there's a benevolent god watching over us the resulting polynomial will approximate the blue curve very well there now because we're dealing with a simple polynomial it's a no-brainer to calculate the area under the pink polynomial curve exactly using standard calculus and if the actual blue function we're interested in is not too wild and there are theorems saying exactly what it means then the area under the pink polynomial will be an amazingly good approximation to the true area then finally we just have to account for the original scaling and we're done also if we are willing to do the careful work and if we shrink the small step size let the step size tend to zero then we can derive the whole of school calculus from the calculus of sequences obviously in this last part i've glossed over plenty of details and ignored all the nasty things that can go wrong but that's plenty of details for today i think definitely great stuff don't you think and after seeing all this calculus will never again be the same for you right okay to finish off let me sketch a proof of the gregorian newton formula for you the proof is very pretty and surprisingly easy quite a few more hormones ahead for the intrepid amongst you ready to get your hands dirty well in usual mathologer style i'll focus on showing why the formula works for a sufficiently general example once you see how the formula works for this example you should be fine to understand why it works in general but always works so there's our sample sequence and the row of differences and so why does the formula at the bottom spit out the sample sequence on top well to explain this let's first make up the difference scheme for n choose 4. remember that for us n choose 4 is just an abbreviation for this expression now plugging in n is equal to 0 obviously makes the expression zero right and the same is true if we plug in n is equal to one two and three this means that the first four terms of our n choose four sequence are zeros now zero minus zero is zero and so differencing those first zeros gives a triangle of zeros what's the fifth term of the sequence well that's four choose four which is equal to one now one minus zero zero one 1 0 again and again and again now because n choose 4 is a degree 4 polynomial we know that the fifth row of the difference scheme is a constant but since the first entry of this row is one it's all ones in this row right pretty pretty and that means that at this point we can hit the algebra autopilot button to reconstruct the rest of the difference scheme let's just see what happens [Music] do those numbers look familiar no what about if we rotate better can you see it now of course that's pascal's triangle and that's really no surprise the way we reconstructed the different scheme by adding up from the ones is exactly the way pascal's triangle grows downwards one plus one is two three plus three is six and so on okay let's rotate back what's really important for the way this works is that the bottom constant rows are ones and the fact that all entries on the first diagonal are zeros this pins down the whole different scheme okay now that was n choose four what about n choose three well it turns out that we can get n choose three simply by zapping the first row of n choose four fantastic of course this can be expressed by saying that the differences of n choose four gives the scheme for n choose three of course this generalizes getting there promise again a different scheme that ends in a constant row is completely determined by that constant row together with the first entries of its rows for example that scheme there has just one row and so is completely determined by the one in front and here the diagonals of the first entries that pin down all the other binomial coefficient sequences now let's build the gregor newton formula for our sample sequence so what happens to these diagonals on the left when we multiply them by the highlighted numbers well the zeros stay zeros and the ones turn into those numbers right there there now we add up in terms of the diagonals that's just superimposing and adding that shows that the binomial sum on the left gives rise to a different scheme that has exactly the same leading entries as that of our sample sequence but since the leading entries pin down the whole different scheme the different schemes of both must be the same tada proved how amazingly pretty was that well in the first instance this all only amounts to a proof that gregory neutral works for sequences whose different schemes fizzle out in a constant role like our mystery sequence what about sequences whose different schemes don't fizzle out well actually it's very easy to extend what we just said to a complete proof covering all sequences again for those of you who have proofs for breakfast can you add the missing details hint remember the part where i said two minus two is equal to zero and now since you are all sequence grandmasters that's all for today we really did cover a lot of ground and if you've made it all the way to here you've earned my seal of approval hope you enjoyed the video until next time [Music] [Music] [Music] [Music] [Music] you

Channel: Mathologer

Views: 204,449

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Id: 4AuV93LOPcE

Length: 47min 10sec (2830 seconds)

Published: Sat Oct 02 2021