5

44 Prove that(m m (-1)"1 ~n2 +n3 =13 n3 ntnz+n3=nwhere the summation extends over all nonnegative integral solutions of n +nz + n3 =n...

Question

44 Prove that(m m (-1)"1 ~n2 +n3 =13 n3 ntnz+n3=nwhere the summation extends over all nonnegative integral solutions of n +nz + n3 =n

44 Prove that (m m (-1)"1 ~n2 +n3 =13 n3 ntnz+n3=n where the summation extends over all nonnegative integral solutions of n +nz + n3 =n



Answers

Prove that $A(m+1, n) \geq A(m, n)$ whenever $m$ and $n$ are nonnegative integers.

Were asked to prove that a of M and plus warn is greater than a of mn where m and and are non negative integers and a is the Ackerman function. So you recall that their immune function of M N is defined by to end and equal zero zero. If m equals one, then equals zero Sorry, that is greater than one and equal zero to it is greater than equal one and n equals one and a of M minus one a. M and minus one. Them is greater than or equal to one and is greater than or equal to two. So to prove this statement, we're going to have to use induction. Excuse me. Let PNB the statement A of m n plus one is greater than a M and for role non negative integers m And for the base case we have and equal zero first in this case we have that a M one is equal to in this case is going to be too just greater than zero, which is equal to a of M zero. So checks out here and now for another base case we have an is equal to one and a of I M. And one plus one is two he's going to be equal to by the recursive definition. Well, it kind actually depends on what M is. Uh, but really, this is equal to a of M minus one. Well, I guess I'll break this up on the cases. So if em is equal to zero, then it follows that this is going to be two times in, which is two times two, which is four, and it's equal to zero. And then this is going to be a of M minus one. A of actually recall that earlier he proved a statement saying that for n greater than equal one, we have that a of em to is equal to four. So it follows that in general, this is just equal to four, which is going to be greater than to which is equal to a of em one. So the statement also checks out. So a proven the base case now for the inductive step. Suppose that PK is true for some k greater than or equal to one. Then we have that It's implies that a M K plus one is greater than a of M kay for all m greater than or equal to zero. So you need to prove that P f K plus one is true. So to prove that statement, we're going to let em be some non negative integer. And we have that a of i m. And then K Plus two is going to be equal to buy the recursive definition a then minus one. And this is going to be valid because it's almost m is 94 0 All right, to break this up into two cases. Really? Yes, an equal zero. And we have that A of EM K Plus two is when a of zero K plus two, which by definition is two times K plus two, which is clearly greater than two times K plus one, which is equal to a of zero K plus one. It works out in that case. Now I m is greater than or equal to one, and we have that A of M K plus two by the recursive definition is going to be a M minus one a. M. K plus one, and recall that by an inductive hypothesis we have a of M K plus one is greater than a of M kay. And so, since this is an increasing function, this is going to be greater than increasing sequence. I should say a of M minus one. This is greater than a of I m. Okay, and this is equal to by the recursive definition or gets created just equivalently as if m minus one a of EM K plus one minus one. So this way it's easier to see that this is equivalent to a of EM K plus one, which is we want to show. So it follows that the statement keep k plus one is true, therefore, by the principal of mathematical induction it follows. That's PM is true for all positive injures end which really means that a of em and plus one is strictly greater than if i m and for all non negative integers m and for all non negative integers and

Everyone. Today we're solving another problem from chapter nine. Section four of the textbook in which we are dealing with not a mathematical induction on before me given is that's okay, equals, Yeah, three last seven also happen. Lost 15 plus the minute the doctor worked out between the continuation of it and then they add, or and minus one crime disease. And that's all set equal toe end times two and plus one in practice, ese. So it's quite a lot of information, and in order to appropriately use mathematical induction, um, your first step you're taking this problem is to crew that when n equals wants All right, number one here. When an equals one, you have to prove that this would be equal to that. So this would be for trying is one minus one. We're trying to see if that equals one would be two times one again. We're trying to see if this equals this. So it would be one times. Do you guys want just to us? What? So I'm the right side of the equation. Two plus one is three times one or simply three. And then on the left side of the equation. You have four times one minus one, which is four minus one, which is also so your first part is done. All right. Now moving on to our second port, which is definitely more complicated. Um, the second part actually involves two parts. So the first part is too. I assume that you can have s o k equals acid end. Um, for any interest, your K on and then the second part is to show that s stuck A plus one, um, equals a certain in value equals. Basically, this with K plus one substituted it. So what that looks like in writing for s a case of one s sub k plus one is you would have asked South K plus one again. That's gonna equal all of this. So I'll just write ASAP. Okay, If you wanted to, you could always write it out, and then you're gonna add this. But again, you're gonna substitute a plus one for this. And you're gonna also substitute a plus one in for us. So what, Kate plus one looks like on the left side of the equation is that you have four times a close one minus one that equals us. Okay. Um what? Plus four times K. I'm distributing that 44 times case for Kay four times one is force of four K plus four miles. One is plus four K minus three. And just so you are aware this initially is set the nice humans. That's okay. And then I'd have to see if ASA People's One is equal s suck A 100 k plus one for this used to right here, right now that we have this, um, you know that the value of your s f k has to be equal to this right here so we can plug in for that. And again, we're assuming that and people came for any interest. Okay? It could be equal to K. So substituting in you, it gets okay. Times Do you hate less? One plus or K? Oh, and by the way, I made a mistake. When you do one time swore, it's positive for, and then you subtract one. It's positively plus three right there. Let's for And then let's work a busted to distribute that kay That's gonna equal what I'm doing. I was distributing. OK, so you got Tuesday square and then it would be adding a K. And then you could combine that, plus que with this plus four k to get plus five. Okay, and then you just have the custody. And now we're trying to see if this is thesafeside thing as que plus one times two K plus three you're trying to see it does to K square plus five K plus three equal the right side of the equation when we have a plus one. Because you already substituted in K plus one on the left side equation. That's where he got that four K plus three. So on the right side, you would have again K plus one times two times K plus one. And then you would also have. So you have cable is one times to keep us one. And then you close parentheses, just going to erase that, Um, and then when you distribute all of this out, you would get so on the left side of the equation again, you would have the same thing, but on the right side, it would equal Oops. Um, so this would be K plus one times two K plus two. So you would get to U K squared. And that guy did Kate times two k that I do Kate Times too, which is two k. Then I would do, um, kay times want to k times once I get plus four k plus to you. Oh, I forgot that. Plus one, that's why. Okay. So again, I substituted in K for and and then I substituted K plus one in for cake. So those were the two stops you took in the second part. But when I substitute in K plus one for and eventually I forgot this last one here. So I need to add that plus one. I was wondering, like, why is it not be equal? So this would be to pay plus three. Yeah. Sorry about this, guys. Ah, okay. This is Pete UK. Last three. Yeah, And then to pay times K S t o k squared. Like I said. And then you do plus three. Hey, Plus two k. So this would be actually plus five k on your right side of the equation. It's like a what I did. There's K times three and then two k times one and then I do, plus three times one which is plus de and hopefully noticed that that is equal to that. So thus when we combine parts one and parts to using mathematical induction, that is how you solve this problem so we can put a check there. We've got two checks and we've proved this formula s O. If you guys enjoy this video, please at the like right next to it. I hope you'll have a nice day. And thanks for watching.

OK, so first we have the anchor. That is for any close to one. We can got one plus one over one Reach eco's to hear and ecos one. So because who on blast one you consume too. That is one Aneke owes to one. So now we come to the second staff That is assumed this equation is true for any Kosuke. So for any coast. Okay, we assume one plus one over one times one plus well over two times one glass. Well, okay. You closed too. Okay. Plus one. So the last type is Thio. Consider when any Kosuke plus one suffers. We can have all this turn like one. Lost one over. When? Times one plus well over. Okay, and now we have one plus one over. Okay, plus one. So as we have assumed that when he goes okay, this for this woman Rico's. Okay, plus one. So this one you Cho soo cope. Last one. So now what? We got ISS. Okay, last one times. Okay. Across one. Okay. Plus one plus one. And here we can see this woman. Orie coast. Okay. Manners too. And though so ecos who and class one So against this formula is valid for all the integral in teacher in

So the first step you should try calling any Yukos one. So the formula on the life you just because tow the some off I from 1 to 1 and I to the power of full, you can see that dressy coast one end up from there on the right. ECOSOC 30 and up here is one times unp last one just one plus one. So there's two and also two times one plus one that history. And also the last term is three times one Squire plus rate hams one minus one, which is five. So we can see the from there on variety. Also, he coast one So they are the same. So from their house, when any coast one. So in the second stab, you assume that the formula also holds When on the coast. Okay, that is Yeah. Sorry. That is the some off I from one to k and I to the power off youcause who waas on the right is 30. Up here is okay times. Okay. Plus one, too. Okay, Low slung Onda three. Okay, Squire plus three k my nose one. So the in the last time we track when Andi calls to K plus one. So the left term should be There's some off when I from one to the cape last one I to the power Awful reach dress. Nico too. These terms, lass. Okay. Plus one to the power off full. Because this, um just this term plus que plus one too powerful. So we just check that in and we can see Here is a 30 also he found the God. Same term here, So we all have Okay. Plus one all side on years 30. Okay. Plus one to the power of three plus okay Times too. Okay. Plus one times three k Squire plus three K minus one. And also we can see this term Ah que plus one to the power of three dress ecos who okay to the power of three plus six k Squire plus six pay plus one. So substitute this term into this function, we can see the formula on the life dressy clothes. Who? Okay, plus one terms. Okay. Plus two terms to K plus three and those. So if we make the guy there were, huh? This one in this racket plus three k plus one. My nose one. That is just what we should get from the farmer on the right so we can see this woman are still holds when a Nikos who k plus one.


Similar Solved Questions

3 answers
~Jed Xes ~H#s p~ h 48M-: 0 052)7 Lysd01 Th ( 'LO[E13o1u! Jo spunoq 341 YIM dn JUeJ nof MOY 3utmoYS suope no[e? 341 e apn[oul nok ams #xeJN 'doo[ Io1no J41 JO ?pISur pue doo[ IJUut 341 ?piSIno 00 831 94 Jo ea1e341 puw u3y1 0 sojz + Z^ =4 UOJEu[ 341 4p1aYSU""( JAsQ
~Jed Xes ~H #s p~ h 48 M-: 0 052)7 Ly sd01 Th ( 'LO[E13o1u! Jo spunoq 341 YIM dn JUeJ nof MOY 3utmoYS suope no[e? 341 e apn[oul nok ams #xeJN 'doo[ Io1no J41 JO ?pISur pue doo[ IJUut 341 ?piSIno 00 831 94 Jo ea1e341 puw u3y1 0 sojz + Z^ =4 UOJEu[ 341 4p1aYS U""( J AsQ...
5 answers
5 e*(2-e)& A3-292 e+3D) 291-2
5 e*(2-e)& A3-2 92 e+3 D) 2 91-2...
5 answers
1. Show below is the mass spectrum for propionic acid. Draw the molecular mass fragments that contribute significantly to m/z = 29, 45 & 74. In addition identify the base peak signal and molecular ion signal on the mass spectrum.10080[ L 4020t+t+++TTT 10 15 20 2530TttttH+ 35 40 45 50 m/z5560657075
1. Show below is the mass spectrum for propionic acid. Draw the molecular mass fragments that contribute significantly to m/z = 29, 45 & 74. In addition identify the base peak signal and molecular ion signal on the mass spectrum. 100 80 [ L 40 20 t+t+++TTT 10 15 20 25 30 TttttH+ 35 40 45 50 m/z...
5 answers
LetA = [-2 2 point) Set up the matrix Apoints) Find the det(A Al ) and identify the characteristic polynomial of A.points) List the eigenvalues of _ along with the algebraie multiplicity of ech eigenvalue:(4 points) Find basis for the eigenspace corresponding t0 the eigenvalue 0. (Only find basis for this eigenspace You do nOt needto do this for the Other cigenvalue(s) )point) State the geomctric multiplicity of the cigenvalue
LetA = [-2 2 point) Set up the matrix A points) Find the det(A Al ) and identify the characteristic polynomial of A. points) List the eigenvalues of _ along with the algebraie multiplicity of ech eigenvalue: (4 points) Find basis for the eigenspace corresponding t0 the eigenvalue 0. (Only find basis...
5 answers
Blood agar smooth; translucent; gray colonies that are 1-2 mmin diameter:Beta hemolyticCatalase positiveMotility umbrella- shapedEsculin = positiveH2S = negativeTreatment combination of ampicillin or penicillin with an aminoglycoside:Identify the Organism
Blood agar smooth; translucent; gray colonies that are 1-2 mmin diameter: Beta hemolytic Catalase positive Motility umbrella- shaped Esculin = positive H2S = negative Treatment combination of ampicillin or penicillin with an aminoglycoside: Identify the Organism...
5 answers
Write the partial fraction decomposition of the rational expression6x2 6x + 3
Write the partial fraction decomposition of the rational expression 6x2 6x + 3...
5 answers
2. Determine an expression for U So the substitution rule can be used t0 evaluate each integral. a) fcos(x)3edx b) f2xe' dx c) Jsin?x cosxdx d) J2r-2x(x I)dx e) ] A+Ldx +3x
2. Determine an expression for U So the substitution rule can be used t0 evaluate each integral. a) fcos(x)3edx b) f2xe' dx c) Jsin?x cosxdx d) J2r-2x(x I)dx e) ] A+Ldx +3x...
5 answers
Evahale:jvS di?Add Flle02; Using Irigsubstitution;Exbale:Add Fi
Evahale: jvS di? Add Flle 02; Using Irigsubstitution; Exbale: Add Fi...
5 answers
Calculate the 2oth percentile of the data shown3.6 8.4 10.3 17 26.7 40.1 45.4 47.1 49.5 49.7 59.1 61.5 66.169.3Submit Question
Calculate the 2oth percentile of the data shown 3.6 8.4 10.3 17 26.7 40.1 45.4 47.1 49.5 49.7 59.1 61.5 66.1 69.3 Submit Question...
5 answers
16_The possible products of the following reaction are:BrCH;OHCH3CH3CH3CH3OCH3IVOHA) 1, III B) 1, II, III 9) III, V IL IV, V E) II, IIIOCH;CHzCHg
16_ The possible products of the following reaction are: Br CH;OH CH3 CH3 CH3 CH3OCH3 IV OH A) 1, III B) 1, II, III 9) III, V IL IV, V E) II, III OCH; CHz CHg...
4 answers
Use the improved Euler'$ method to obtain four-decimal approximation of the indicated value_ First use use 0.05_0.1 and thenY' = 2x - Jy + 3, Y(1) = 6; Y(1.5)h = 0.1 Y(1.5) ~ h = 0.05 Y( 1.5) ~
Use the improved Euler'$ method to obtain four-decimal approximation of the indicated value_ First use use 0.05_ 0.1 and then Y' = 2x - Jy + 3, Y(1) = 6; Y(1.5) h = 0.1 Y(1.5) ~ h = 0.05 Y( 1.5) ~...
1 answers
(a) $\mathrm{A} 0.140$ -kg baseball, pitched at $40.0 \mathrm{m} / \mathrm{s}$ horizontally and perpendicular to the Earth's horizontal $5.00 \times 10^{-5} \mathrm{T}$ field, has a 100 -nC charge on it. What distance is it deflected from its path by the magnetic force, after traveling $30.0 \mathrm{m}$ horizontally? (b) Would you suggest this as a secret technique for a pitcher to throw curve balls?
(a) $\mathrm{A} 0.140$ -kg baseball, pitched at $40.0 \mathrm{m} / \mathrm{s}$ horizontally and perpendicular to the Earth's horizontal $5.00 \times 10^{-5} \mathrm{T}$ field, has a 100 -nC charge on it. What distance is it deflected from its path by the magnetic force, after traveling $30.0 \m...
5 answers
Question 4 (1 point) ListenWhat is thc IUPAC namc o the following compound?tran ~isopfopy] ~methykcyclopentanccis-]-1sopropyl 2 methr-Icyclopentanecis-]-tert-butyl-2-methy-IcyclopentaneWton~tert-buty] - -methylcyclop-ntaneQuestion 5 (1 point)LatenWhai 15 the approximate dihedral anglc bclwec [ne tWro chlonle Tdndichloro-yclobcxanc120"180"
Question 4 (1 point) Listen What is thc IUPAC namc o the following compound? tran ~isopfopy] ~methykcyclopentanc cis-]-1sopropyl 2 methr-Icyclopentane cis-]-tert-butyl-2-methy-Icyclopentane Wton ~tert-buty] - -methylcyclop-ntane Question 5 (1 point) Laten Whai 15 the approximate dihedral anglc bclwe...
5 answers
Find the area of the surface generated by revolving the curve0sxs2, aboul the x-axis_Set up the integral that gives the area of the given surface(Type exact answers; using - as needed )
Find the area of the surface generated by revolving the curve 0sxs2, aboul the x-axis_ Set up the integral that gives the area of the given surface (Type exact answers; using - as needed )...
5 answers
-22 % 42 $ L { 1 2 { { 8 F 2 }2
-22 % 42 $ L { 1 2 { { 8 F 2 } 2...
5 answers
[0r2 PoinylDCAI 5FAFLAalerMhe CenonAear (f4esi2] W Honee n( {ne derunnale IE W m St Rht numeralar be poNronisa NIIJ Cegn * CCz luts tanncceelnt FfNmT
[0r2 Poinyl DCAI 5 FAFL Aaler Mhe CenonAear (f4esi2] W Honee n( {ne derunnale IE W m St Rht numeralar be poNronisa NIIJ Cegn * CCz luts tan ncceelnt FfNmT...
5 answers
CNeutPlcast complceIehlcbelonAalitFomullKanhonOridoHydrogen phosphateRicuchonalcLOmLLLEidhnPerchloratenttePhosphateactually exist naturally but can DISCLAIMER:Some of the compounds generated by this website do no discussed in this coursc. For this rcason, I5 not : good idca t0 be named Fusing the nomenclature mules the answcrs on (doing s0 will not help you--and it is also academic dishonesty )Aniun
CNeut Plcast complce Iehlcbelon Aalit Fomull Kanhon Orido Hydrogen phosphate Ricuchonalc LOmLLL Eidhn Perchlorate ntte Phosphate actually exist naturally but can DISCLAIMER:Some of the compounds generated by this website do no discussed in this coursc. For this rcason, I5 not : good idca t0 be named...
5 answers
[6 pointe] Perform the following conversions and show allyour work for fufl credit. You may need look up conversion factors in yourbook online . 13,500 mg of NaCl to grams NaCl 25 8 0f He to moles of Ile 0.3265 moles of CH_COOH grams of CH;COOH 1.5 g of water ML of water
[6 pointe] Perform the following conversions and show allyour work for fufl credit. You may need look up conversion factors in yourbook online . 13,500 mg of NaCl to grams NaCl 25 8 0f He to moles of Ile 0.3265 moles of CH_COOH grams of CH;COOH 1.5 g of water ML of water...

-- 0.021449--