5

Prool secton choice) for 13 points induction proof; either #17 OR #18 (circle your Do an ~n-12 @DProve by induction (VneN) 6]/ #' and 7 cents as follows proble...

Question

Prool secton choice) for 13 points induction proof; either #17 OR #18 (circle your Do an ~n-12 @DProve by induction (VneN) 6]/ #' and 7 cents as follows problem for stamps worth 4 cents etamn for integers a,b20 _

Prool secton choice) for 13 points induction proof; either #17 OR #18 (circle your Do an ~n-12 @DProve by induction (VneN) 6]/ #' and 7 cents as follows problem for stamps worth 4 cents etamn for integers a,b20 _



Answers

Let $P(n)$ be the statement that a postage of $n$ cents can be formed using just 4 -cent stamps and 7 -cent stamps. The parts of this exercise outline a strong induction proof that $P(n)$ is true for all integers $n \geq 18$ .
a) Show that the statements $P(18), P(19), P(20),$ and
$P(21)$ are true, completing the basis step of a proof by
strong induction that $P(n)$ is true for all integers $n \geq 18$ .
b) What is the inductive hypothesis of a proof by strong
induction that $P(n)$ is true for all integers $n \geq 18 ?$
c) What do you need to prove in the inductive step of a
proof that $P(n)$ is true for all integers $n \geq 18 ?$
d) Complete the inductive step for $k \geq 21$ .
e) Explain why these steps show that $P(n)$ is true for all
integers $n \geq 18$ .

High in this question, we are asked to prove using strong induction The statement that Apo stage off incense can be form you sing only three and five cents stems when this dead men with in is at least it. So let's do this first basic step here I do do not only one but three statement for basics that you will see why in a minute. But is easy to see that these are all true, right? We can form Ed nine and 10 sitting posted with only tree and fire since then. Ah nde The reason I do this is that when we have it night in as abyss, we can prove inactives. They're very easily because this Kovar tree concert pretty in danger and so any any other indigent after this can be creep can be created using this this 1st 3 plus pre something So so tree I right, it it it doesn't matter how far we go. We can't just keep increasing tree at the time from one off the trophy starting position and that's how we gonna show our inductive step. So supposed the statement is true for Kay from Ed night in two in for some positive Dejan Then we want to show that is true Foreign plus one indeed we can say that in plus one must be in one off the one of these three kisses, right? Any positive in Hegel? Must be Congress to 01 or two Modelo tree. All right, I write it like this so that it coincide with at night in but they put like the order doesn't matter. So when is Khan granted to model a tree then is off the form eight plus three something like I said earlier when he's divisible by tree so is Ni Plus two is something when is congruent to one, then is 10 plus three something So in all cases we can create that exact posted using the starting three blocks that we have shown that we can create than at tree stands over Until we get that. And so this this proves our in that tape step And so together with best extent, we have shown that Sorry the statement is true for in Greta equal to it. That is it. Thank you

In this problem. In part, they were asked to determine which months of postage can be formed. Using just four sent an 11 cent stamps so going out to use four cents an 11 cent stamps, of course, weaken make four cents of postage. You can also make two times for his eight cents postage. He can also make 11 cents a postage. We could make three times four is 12 cents. A postage should make four times forward 16 cents of postage you could make five times for, which is 20 cents a postage. And we could make 11 times to which is 22 cents of postage. You can also make six times four, which is 24 cents a postage. We can make seven times for just 28 cents a postage. We could make eight times forward, just 32 cents postage. You could make three times 11 which is 33 cents of postage, I said go left up some options here So that 48 11 12 then also we have 15 is an option. It's 11 plus four and 16 and of course we have 15 plus four. 19 is an option. Then 20 and then we have that 19 plus forward. Just 23 is an option and we have 22 plus four, which is 26 is an option and 23 plus forwards is 27 is an option. Course, 28 is an option, and we have that 30 is an option. This is 26 plus four and we have the 31 which is 27 plus four is also an option. And then we see that 30 plus forward just 34 is an option. And so this just goes on increments of what interest you're from that on and in part B. Whereas to prove the answer to part, they used the principle of mathematical induction So more to say that the statement PN is that hostage of end sense can be formed using just for sent and 11 sent stamps. Then the statement is that PN It's true furrow and greater than or equal to 30. And to prove this using mathematical induction so we have clearly shown before basis. Case P 30. In this case, 30 is equal to two times 11 which is 22 plus two times for which is a. So it's clearly true. And we're going to suppose as our inductive hypothesis that p que is true for some que greater than or equal to 30. I want to show that p k plus one is true. So p K plus one. This is the statement that K plus one could be formed using just 11 4 as multiples. So we have that K plus one is equal to Yeah, So now let's consider some cases. So let's suppose that que cent stamp could be formed using only multiples of 11. And it follows that the K plus one cent stamp me formed using l minus one. Time is 11 plus three times for which is 12 because, well, 11 times l is K and then they get negative. 11 plus 12 is one. So it proven the statement is true in this case. Now, suppose that K is equal to some multiple of four. Well, then we have that K plus one is going to be equal to l minus. In this case, we have that 11 is going to be one less than 12. So you see one of you in the way to think about this is that since que is greater than or equal to 30 So it follows that que must actually greater than they were 32. If it's multiple of four and it follows that l must be greater than or equal to eight now, who'd replace 84 cents stance with 3 11 since stamps, So we have that K plus one is equal to l minus eight times four. So we have four times l is K and then negative eight times four is negative 32 plus three times 11 which is 33 So shown. The statement is also true in this case. Finally, since you've shown that PK plus one is true in both cases, you shown that so PK plus one is true in general and therefore by the principal of mathematical induction. We have that p n is true for all and greater than or equal to 30 and in part, C were asked to prove the answer to a again this time using strong induction, and were asked to point out the difference between the induction hypothesis and this proof from that of the hypothesis from the proof in part B so in part c. Well, we have that statement p 30 p 31 he 32 and p 33 are all true from party and this is going to be our basis step. And so, for the next hypothesis, let's suppose that statements p 30 p 31 all the way up through PK are true for some K greater than or equal to 33. We want to show that he K plus one is also true. So this means that PK minus three is true and so it follows that we confined. It's, um integers a and B such that K minus three equals eight times four plus b times 11 And so we have that K plus one is equal to a plus one times for plus B times 11 and since a plus one is also impure bees and interest your it follows that PK plus one is true. And so here the difference in the productive hypothesis was getting assuming simply that PK was true. You assume that all statements p 30 up through PK are true. And now since we've shown that PK plus one is true, it follows by the principal of strong induction statement. PN is true parole and great early quarter 30. So even though

So he has a bullying function. P. M. N. Is equal to four power in minus one. And you have that this is divisible by three minutes to prove it with an induction. So let's see spl one that is uh for to the power of Guan which is 4 -1. That is three NPR zero. In case they care This case is going to be 4 to policy which is 1 -1. That's going to be zero. So they're both divisible by two week. Good. Now let's say P F K. Which is a call support to the power of K -1 is invisible by three. Then we need to prove the same for P R. K plus one. So ah PRK MPO Okay plus one minus P F K is going to be mhm For To the power of K-plus one -1 -4 to the power. Okay. Was the one that's gonna give us for to the power of K times or minus one. Do they call to three times or to recover? Okay, so the difference between P F K plus one and PRK is divisible by three. Therefore If you are Cape Close one is equal to P F K. I'm just kind of say here divisible by three. He's been called Sapir cables on his P. F. K plus three times. For to the power of K Which is also divisible by three then. Yeah. OK Gloves one is There was a goal by three. So we have proven our scenery

In this question we need to prove the following that if we have a function p event is nine to the n minus one. That is divisible by four minutes. Use induction for it. Based on the induction. Let's prove our base case. Pl one Yeah one will be 9 to the power of 1 -1 which is a 4-8. And I confirmed that that is the with the Mobile four. Now let's say hypothetically that Okay Which is 9 to the power of K -1. Yes, it was about by poor. We need to prove that P M. K plus one will be which is going to which will be um 92 decades Plus 1 -1 is divisible by four. Even though by four. I'm sorry if I said 3-4. So what is P F K plus one minus P. R. K. So P L. K plus farm minus P. F. K. It's gonna be nine. The power of K plus one minus 19. Power. Okay? Or if I packed her the following. Thank you. Problem. K times Um 9 -1 or eight. So This is divisible by four. And if PFK plus one is equal to 9 to the power of K -1 plus eight times 90. Power. OK And if we have that decision was over about four and a prostitute double by four. Then that proves it that P. F. K plus one is also divisible. And so with the induction we have proven what was required


Similar Solved Questions

3 answers
An ODE / f(w) where the ten f(v) can cpleSU fuction of the ratio OMly. Mid homogencous Such eqpatious CAI awzys be tratusfonued JO spatrable (puations bw cutulge of the variabk.
An ODE / f(w) where the ten f(v) can cpleSU fuction of the ratio OMly. Mid homogencous Such eqpatious CAI awzys be tratusfonued JO spatrable (puations bw cutulge of the variabk....
5 answers
Evaluate the following integral over the Region R(Answer accurate to 2 decimal places).2 2 f I 66 - - y _ dA R3R = {(7,0) | 0 < r < 3,r < 0 < 4"}
Evaluate the following integral over the Region R (Answer accurate to 2 decimal places). 2 2 f I 66 - - y _ dA R 3 R = {(7,0) | 0 < r < 3,r < 0 < 4"}...
5 answers
Problem 2_ For €R; we define the set Sa {1 €R:r < a 25} Show that 0 if and only ifa € (-5,5)
Problem 2_ For €R; we define the set Sa {1 €R:r < a 25} Show that 0 if and only ifa € (-5,5)...
5 answers
Finc Ihe arc Ienglh parameter along Ino curve Irorn Ine poini whierebv avaluating the Integralvlc) dt Tuen' fndiheIengtn of the Indicaled portion 0f Ihe curve 5f =(2 300 - (0 + 4+ (4 Jk 19120
Finc Ihe arc Ienglh parameter along Ino curve Irorn Ine poini whiere bv avaluating the Integral vlc) dt Tuen' fndihe Iengtn of the Indicaled portion 0f Ihe curve 5f =(2 300 - (0 + 4+ (4 Jk 19120...
5 answers
If M.D. is 12, the value of S.D. will be(A) 15(B) 12(C) 24(D) None of these
If M.D. is 12, the value of S.D. will be (A) 15 (B) 12 (C) 24 (D) None of these...
5 answers
Translate each statement into an inequality. Use $x$ as the variable.Chicago received more than 5 in. of snow.
Translate each statement into an inequality. Use $x$ as the variable. Chicago received more than 5 in. of snow....
5 answers
Solve the initial-value problem shown below: dy dc =I V1-v? y (0) = 0 Give an exact formula for ysin (a)3sin3
Solve the initial-value problem shown below: dy dc =I V1-v? y (0) = 0 Give an exact formula for y sin (a) 3 sin 3...
1 answers
Indicate the direction of the important bond moments in each of the following compounds (neglect $\mathrm{C}-\mathrm{H}$ bonds). You should also give the direction of the net dipole moment for the molecule. If there is no net dipole moment, state that $\mu=0$ D. (a) cis-CHF =CHF $\quad$ (b) trans-CHF = CHF $\quad$ (c) $\mathrm{CH}_{2}=\mathrm{CF}_{2}$ (d) $\mathrm{CF}_{2}=\mathrm{CF}_{2}$
Indicate the direction of the important bond moments in each of the following compounds (neglect $\mathrm{C}-\mathrm{H}$ bonds). You should also give the direction of the net dipole moment for the molecule. If there is no net dipole moment, state that $\mu=0$ D. (a) cis-CHF =CHF $\quad$ (b) trans-CH...
5 answers
QuestionDNA what kind of base pairing can occur? (CHOOSE ALL THAT APPLY)Questlon 2Hydrogen bonds play the following rolefs): (CHOOSE ALL THAT APPLY)Qquestlon 3Which of the following are the Main functions protcinsbodies?QuestionWhat are the main sources of eneigy -anima ceilstQescieuWynlct dementc are commonnaund assocuuteo wich carbonliving rganisms? CHOOSE ALL THAT APPLY
Question DNA what kind of base pairing can occur? (CHOOSE ALL THAT APPLY) Questlon 2 Hydrogen bonds play the following rolefs): (CHOOSE ALL THAT APPLY) Qquestlon 3 Which of the following are the Main functions protcins bodies? Question What are the main sources of eneigy - anima ceilst Qescieu Wynlc...
5 answers
21. Determine consumers' surplus (CS) and producers' surplus (PS) under market: equilibrium 0 Demand: p = 400-92.1 Supplv: p = 20q + 100
21. Determine consumers' surplus (CS) and producers' surplus (PS) under market: equilibrium 0 Demand: p = 400-92.1 Supplv: p = 20q + 100...
5 answers
Write English phrase as an algebraic expression. Then simplify the expression. Let x represent the number.The difference between the product of six and a number and negative two times the number
Write English phrase as an algebraic expression. Then simplify the expression. Let x represent the number. The difference between the product of six and a number and negative two times the number...
5 answers
In each case, a linear relationship between two quantities is described. If the relationship were graphed, what would be the slope of the line?a. The sales of new cars increased by 15 every 2 months.b. There were 35 fewer robberies for each dozen police officers added to the force.c. One acre of forest is being destroyed every 30 seconds.
In each case, a linear relationship between two quantities is described. If the relationship were graphed, what would be the slope of the line? a. The sales of new cars increased by 15 every 2 months. b. There were 35 fewer robberies for each dozen police officers added to the force. c. One acre of ...
1 answers
Library of Parent Functions In Exercises $105-107$ , deter- mine which polynomial function(s) may be represented by the graph shown. There may be more than one correct answer. $$ \begin{array}{l}{\text { (a) } f(x)=(x-1)^{2}(x+2)^{2}} \\ {\text { (b) } f(x)=(x-1)(x+2)} \\ {\text { (c) } f(x)=(x+1)^{2}(x-2)^{2}} \\ {\text { (d) } f(x)=-(x-1)^{2}(x+2)^{2}} \\ {\text { (e) } f(x)=-(x+1)^{2}(x-2)^{2}}\end{array} $$
Library of Parent Functions In Exercises $105-107$ , deter- mine which polynomial function(s) may be represented by the graph shown. There may be more than one correct answer. $$ \begin{array}{l}{\text { (a) } f(x)=(x-1)^{2}(x+2)^{2}} \\ {\text { (b) } f(x)=(x-1)(x+2)} \\ {\text { (c) } f(x)=(x+1...
5 answers
8 limp that What Poot Falr I} The background? Degree? the the the S50d 1 1 Question manager WW The onlv Each N that a that nunjgCT < educationzi U Tool re managei HH 1 1 whal 01 L 1 E Ydsus0 1 the 8fp L probability i4l hoted Wu 66 3 1 1 that this or PhD? 1 Codbge ) 1 manaeennas that manager 1 I}
8 limp that What Poot Falr I} The background? Degree? the the the S50d 1 1 Question manager WW The onlv Each N that a that nunjgCT < educationzi U Tool re managei HH 1 1 whal 01 L 1 E Ydsus0 1 the 8fp L probability i4l hoted Wu 66 3 1 1 that this or PhD? 1 Codbge ) 1 manaeennas that manager 1 I...
5 answers
[2r3 Potnts]PREvIOUSA eksDEVORESTATI ?.E.019.MLsHOTFSRoukaachmniGIc ANoihlWrubURqupnrd +#ht? Jdigiret 07t orinmanaUSt SaLTCuculeta tho Iantue ongugatu H4
[2r3 Potnts] PREvIOUSA eks DEVORESTATI ?.E.019.MLs HOTFS Roukaachm niGIc ANoihl WrubU Rqupnrd +#ht? Jdigiret 07t orinmana USt SaLT Cuculeta tho I antue on gugatu H4...
5 answers
Consider the polar function: "(o)sin( 0). sin(0)0 < 0 < # (0), { <0 <(a) Graph ,(0) (b) Compute the exact value of the area enclosed by the curve
Consider the polar function: "(o) sin( 0). sin(0) 0 < 0 < # (0), { <0 < (a) Graph ,(0) (b) Compute the exact value of the area enclosed by the curve...
5 answers
Lety =5H: and Yz Find the dislance from lo Ihe subspaco Wol R" spannod by %4 &d Yz' given Ihat the clososl point toyin WisThe distance is (Simplify your ansiver Type an exact answer; using radicals as needed )
Lety = 5H: and Yz Find the dislance from lo Ihe subspaco Wol R" spannod by %4 &d Yz' given Ihat the clososl point toyin Wis The distance is (Simplify your ansiver Type an exact answer; using radicals as needed )...
5 answers
QuestION 11 Kz3a 0.82, then the value ofa 3.0.1031 b. 0.4122 0.6183 0.0.0687
QuestION 11 Kz3a 0.82, then the value ofa 3.0.1031 b. 0.4122 0.6183 0.0.0687...

-- 0.020631--