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Homework 5 Complete each of the following problems Be sure to include any relevant work and explanation. 1. (2 points) Prove by induction that the following equatio...

Question

Homework 5 Complete each of the following problems Be sure to include any relevant work and explanation. 1. (2 points) Prove by induction that the following equation holds for every positive integer n: Zk? = nlnt WZn+1 k=] 2 Let M(xy) denote "x Has emailed y", T(xy) denote has textedy", and P(xy) denote "x has called Iy on the phone' where tne the Cartesian product of the set of students in this class with itself: Translate the following into clear English sentence: Vvax

Homework 5 Complete each of the following problems Be sure to include any relevant work and explanation. 1. (2 points) Prove by induction that the following equation holds for every positive integer n: Zk? = nlnt WZn+1 k=] 2 Let M(xy) denote "x Has emailed y", T(xy) denote has textedy", and P(xy) denote "x has called Iy on the phone' where tne the Cartesian product of the set of students in this class with itself: Translate the following into clear English sentence: Vvax(M(x v)v Tlx V)v Ax,v)) b. Use the given propositiona functions, quantifiers, and any other necessary logical operators to express the following stateme spoken on the phone: 3. (2 points) Prove by induction that the sequence of the positive powers of 3 Is increasing: Homework 6 Meraitatel NAtt



Answers

In Exercises $1-24,$ prove each statement using mathematical induction for all positive integers $n$. $$5^{n} < 5^{n+1}$$

High in this question. We are again us. What's wrong with the proof? They give us the proof. You see induction off this statement p of in saying that Eve X and Y are positive. Vintage is such that the maximum he's in then X equal to y visible. Um, using Just come on, send. I'm sure we all understand that this is not true, right? You can have because it doesn't have anything to do with X and Y being equal. My Maxie Max off to number being some vintages and then faucet to be equal. It's just it just not work. But they give us approve anyway. And us, what's wrong with it? So I will show one off the mistake that can be found. Um, the key here is that X and Y are posted deviant ages. So they show basic step, which is, uh, quite obvious. But then, in the inductive step, that document is that supposed we have posted TV nature extended wide such that the Maxie's now in plus one, they are cuter than the max off X minus one And why my next one must be just in right. And that is true And then they said we can use inductive hypothesis now because now the max is is in, which is what we assume and so inductive about the siege would give these x minus one equals y minutes one. And so they come back to X equal to y, and they use it to prove in the tips that what's wrong is here This too. When we start, we said a X and Y are positive vintages wishes, all right, But then when they use in that TV hypothesis, they using without where If I ng that ace minus one and why minus one Aw, posited in hagia or not. And in some cases they are not. Obviously when x a wise one, right? Just one of them is one then X minus one is seal which is not positive integer anymore, right? And so even though we have inductive hypothesis to use, we must abide by the statement we stayed that we consider extend. Why positive vintages. So he had the use off. Inductive hypotheses can be applied in some kisses because the minus one here may may not be positive vintages anymore. You can be Ciro. And so yeah, the proof breaks now here and that is it. Thank you

Yeah. You see mathematical induction we're going to prove that statement is and it's true for every positive engineer. And we follow these steps here for a verify S. One B. We write this K and C. We write escapist one G. We assume that this case true. And use algebra to change escape to escape was one. And a party we write the conclusion based on steps. A through the statement is in is the following one half plus 1/2 square plus one over two cubed plus up to one over to to the ends of power equals one minus 1/2 to the s power. So we start with very fine statement is one and statement is one has an equality which left side with the left side able to one term one half. That's because the last term here for an equal one is one house. So the sun reduces to only one term which is the first term one half. And the right side is juan minus 1/2 to the one and then we have one half equal one man of minus one half and so on half is equal to. He had two minus one over to that. He's one house when healthy equal one health is true and so is one is true. Now in part B will write statement this case simply replacing and by K. In the general statement here, so we get one half plus 1/2 square. That's one over to Q plus up to one over to to the ends power equal one minus one. Sorry to the cave power. It meant equal one minus 1/2, 2 decades power. And now the statement statement escape is what in part C is one half plus 1/2 square plus one over to cube up to. We know that the last term is one over to to the case was one power but before the term we have this one here is 1/2 decades power and that is equal to one minus one over to the case cables one power which is uh statement we obtained replacing and by cables one here. Now in party we assume that assume that this case true improved that escape. This one is also true. So let's assume that escape is true, is true and there is that these equality is true. One half was 1/2 square. This one over to cube plus up to one over to the case. Power eagle one minus one over to the keith power. So this is what it is true. And now we're going to try to prove that escape was one. This statement here is also true. Okay then if we add the same term both sides of this equation, he had another equality. That is also true. And what we're going to add both size is the term one over to the case plus one. So it's one half plus 1/2 square plus one over to Q plus one hour to to the cave power plus one over to to the case plus one power equal one man is one over to to the case power. There is this term plus the term. We are adding both sides which is this one went over to to the gates plus so we have And these terms both sides of these equality here, which is the same as this. Terrorism equal disturbed and because this we're assuming we're assuming it's true. This must also be true. Okay, so what we get this one half plus 1/2 square plus one over to cube up to one over to to the cave power puts one over to to the k plus one equal one minus. Take an active one common factor and said Francis. Get one over to keith power minus one over to to the cave plus one and so one half plus 1/2 square plus one over. Okay to cube up to one over to to the kate's power plus one over to to the case. This one equal one minus and the common denominator is to to the Caithness one that divided by two. The case power is two times one is two minus one and then 1/2 that's 1/2 square because one or two cube plus up to one over to to the case power plus one over to to the case plus one Power equal one minus two minus one. The numerator we get one over to to the cave plus one. And this is just his statement as K plus one here and so is key was one. Mm Please answer true. Then what we have proved is that statement escape. This one is true whenever sK it's true that for any positive into UK. Mhm. So if we combine a party but we have proved in part A plus this, we have proven party so A plus D plus and that as any is true, statement is in store for every positive interview in. That's because it's one is true following part A and using this for K equals one. We get that? S too must be true because this one is true. And again now knowing that it's too is true for cake was too we know that three is true and we can go on forever and we get this result. That is statement as it is true for every positive in here and

Forgetting the statement to a support Rand is going than in square. They're an escort identical to five. So let's just take any school to five year for us. We get 32 more than 25 vicious fruit I'll just take and physical decay as our induction hypotheses. And we get to raise the bar. Que is where is in case quick this. We have to assume it's true. So let's just look at an induction case Now her and musical two K plus one. We have to prove their civic so we can rewrite this week. Invite this as two ways for K plus one where them tables on hold square. Now we can rewrite that has to raise a parquet into two is good than case where plus two K plus one. Now what you can do is we could use induction hypotheses and we can isolate the two K to a spark A here and we can be right that has two weeks. Market is good in Case Square. We proved that or we assume that full right now Just multiply this holding by two on both sides they get do X work ended. Two is going to then two into K Square. Just case Claire plus K Square. Now they just look at Allegis, make a side note here and see that case Claire. Then Cleaves greater than five is greater than two K plus one. Using deductive reasoning. We can see that since gave national number, it's cooler than five, and that's going to itself. There's no way that two K plus one is ever gonna be great in case where so by that logic we can see that since case goes were than two K plus one. So that means that two K into two square, then to case where rich in itself is well, in case where place to keep us on Regis K plus one whole square. So we can now rewrite this as Tu que institute is greater than K plus one, which is what we had to prove or earlier

This question is from mathematical indication here given four by five plus four way pain square plus poor by faith Q pour to the birth fine in. This is 1 -1 day play in verifies if anyone then yes juan becomes poor by five. is one miners food way five Are the siege four x 5 a minus one. Now this is for five. Well the Saudis him you could. So this is true. No sK the statement. Oh yeah This is four. Where for you plus broadway. Pain square poured away. Five Q plus 455 Okay. It's a call to 1 -1 one better for you. Okay. Mhm. This is a statement tour by faith. Times Square. This is very Q plus pulled away. Okay. Okay. And for way Frank A. one. This is one one bird. five K Plus one. Yeah. No a jamming s caged through. Mhm. Yeah they teach pulled by five. Four way for U squared broadway. Thank you. This is four way Craig. Okay. It's called a one. My nose. One way five. Okay now we are broadway. This is K plus one on border side. Yeah then we have this is for bay. Okay. Plus broadway. Face square. Poor way. Thank you. Four way here for you. Okay. And poor way Play K-plus one Jessica will do one. My nose. This is five K. Plus one message here. One madness one by K. Plus four way Time to the we'll keep this one the suit one minus five K plus one. This is minus five plus four. And here this is one one. Bye. Trying to defer. Okay plus one. I can't if yes K. It's true then. Yeah. Sk plus one is also true. Therefore Yes and this is four x 5. Yes. four x 5 square world way. Thank you. Plus for lead five in the siege one minus one way 5. And it's true for any positive integer. Yeah dissidents. Thank you.


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