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Consider tlute following (corect HGLCILL shichi ilrinols Tennernt iL4T Suppost Thcn n 4 + Jor jome utcar and k 2 + 1 for sofie tnteaet Hene ,nk (2 + 1)(26- 1) = 4l ...

Question

Consider tlute following (corect HGLCILL shichi ilrinols Tennernt iL4T Suppost Thcn n 4 + Jor jome utcar and k 2 + 1 for sofie tnteaet Hene ,nk (2 + 1)(26- 1) = 4l + 24 + 20 + 1. Therefon, nk elaWrite the implication proed by the argument plain Euglish: plain English: E Nwed Write the coutrapositive of the implication argutent? statemeut in proved by the argutent? Write the converse your

Consider tlute following (corect HGLCILL shichi ilrinols Tennernt iL4T Suppost Thcn n 4 + Jor jome utcar and k 2 + 1 for sofie tnteaet Hene ,nk (2 + 1)(26- 1) = 4l + 24 + 20 + 1. Therefon, nk ela Write the implication proed by the argument plain Euglish: plain English: E Nwed Write the coutrapositive of the implication argutent? statemeut in proved by the argutent? Write the converse your



Answers

Determine if each implication is vacuously true for the indicated value of $n$

If $n \geq 4,$ then $2^{n} \geq n^{2} ; n=0,1,2,3$

To show that an implication is Vegas the truth. Um, the implication H implies C is Vegas lee true If it is false. So when you have, can you show that age is false? People say the statement is Vegas lee true. So what we have here is uh, if N is going to be not equal to one. So this is our hypothesis. Um, the conclusion holds when N is equal to zero, so you want to know. Okay, So you can see that We are evaluating the conclusion at zero, but All right. Um, and it's greater than zero is greater than one, but you know that um, zero is not Greater than or equal to one. All right. zero is not good at the reports one. So you would say that our hypothesis coach it's false. So our statement is like do Wesley Sure.

Suppose suppose Absolute value of two X plus two. Less than several points three. So our next in your inequalities are equivalent to so nest inequalities. Inequalities uh huh equivalents equivalent to equivalent? So we have 48 lights AIDS simulates and 1.2. So I have four x plus two, Less than 1.2. So absolute value of four X plus two list uh Less than 1.2. So divide both sides by four. And you have I'm still this value of X plus two, Less than 1.2 divided by four. And this gives us absolute value of X plus two, Less than 3.3. So you realize that what we supposed and this inequality actually words hoods. So then it implies that the state's means this stage names actually is too

This problem. We are going to use mathematical induction to prove that the statement S. N. Is true for every positive into the end so that we follow these steps, verify as one in part a part B will write S. K. A. In percy were right escape this one In party. We assume that escapes true and use algebra to change his K. to escape plus one. The party will write a conclusion based on steps one through the the statement S. N. Is the following two plus four plus eight plus up to To raise to the 10th power equal. Yeah To raise to the templates one power -2. So before we find is one, we can see that this statement we are given can be written this way too. Race to the one plus two square plus two, two, third plus two to the fourth and That up to to raise to the 10th power. That's the sum And the reside is to to the accomplice one. I understood so in this some here this exponents which consists of the successive positive introduce us accounting index because this is the first term and the exponent is one. This is the second term and expanding is to third term is responded to three and so on. It means that we are counting with this opponent the number of terms so we have in general here in terms so we can rephrase the statement as an as saying that the sum of the first And powers of two Is equal to 2 to the amp lists 1 -2. That being said we can start now we're fine as one. In this case we have the some of the first one terms that is there's only one term So there's no some at all but the first time term only so we get to to the one on the left side that is the sun reduced to the first term only. And on the right we have to to the one plus one because We are replacing end by one minus two. So on the left side to the one power is too And on the right side we have to Square -2. That is to equal for -2. That is to equal to This is true because the equality is true. We know that the statement as one is true so we have verified it's one. However, be we are going to right escape. So a statement S. Case is simply this statement as N replacing A nine and by K there is too plus four plus eight plus up to Mhm. To race to the case power equal To to the case Plus one Power -2. And perceived we write this statement escape this one is two plus four plus eight plus. In this case we get the last term will be to to the K plus one power. But we know that before this term we will have to dedicate power. Which is this one here and this is equal to two to the K plus one plus one. And that -2 here. As we can see we have replaced this K here by K plus one here. In fact is K plus two. We have written this way in order to mm clearer have these uh two statements and now in part e we're going to prove that esca escape lies is K plus one. That is if we assume that this case true, we can prove that escape. This one is also true. Yeah that is. We assume sk is true. Yeah. Yes two plus four plus eight plus to to the cage power equal to to the K plus one. Power minus two. And. Mhm. That being true. That is this quality was supposed true. We can add any number of both sides of this equality. And we will get another equality. Which is also true. Must be true. Because we're adding the same expression. Both sides and the separation. We got to add both sides. It's just this term here which is a term we need on the left side to have this some That appears in the left side of escape this one. Yeah. So adding the expression Okay yeah. To keep this one. Both eyes. Yeah. Of the this equality we have. That is true. That two plus four plus eight plus two to the K plus two to the K. Plus one Equal to the gay plus 1 -2 plus To keep this one. Let's see. It's very important to notice this is a term. We are adding both sides and the rest of this depression. This part here and this part here, each other. It's just this equality which we are assuming is true. So because that equality is true and in the same term of size give us a new equality which is also true. Mhm. And so two plus four plus eight plus up to two K plus two to the K plus one is equal to. And now we do some algae around the right side. And we know that we can see that this term and this one is the same. So it's two times that term In that -2. But now here we have a product of two powers of two. Here we have expanded one and here we have expanded K plus one. So we can apply the properties of the product of power with the same base and we will get in that that's equal to two to the one plus K plus one minus two. Because we have left the same base or conservative base. Keep the base and some the exponents that's the property. So we get finally, we can rewrite this as to to the K plus one Plus one is the same thing and this is just Yes Keep This one. Okay. It's important to verify that we have get we had obtained escapist fun and that's the case. The sum up to two K plus one power. Yeah. To the capitalist one and the right side is to to the capitalist one plus one minus two just this year. So we have assumed that this case True. And You see algebra? We proved that escape Lys one must be also true. So sk implies escape this one for any positive integer. Okay, so that means that if we know that this statement is true for some natural number. Okay then with this we have proof here we can uh huh be sure that the statement will be true for the next natural number and now applying again that we know that the statement will be true for the next Natural #2 These last we had and so on so forth. So we can Say that the statement will be true for any natural number if we have at least one or the first we want verified and that has been done in part a verified just as one. So a plane this we know that it's true. It's true, Putting K. one. But knowing now that it's true it's true. We apply again this week cake was too so as two equal is too true will imply that if there is true and so on, so have improved in part A that as one is true and have proved this implication here, we can say that this statement is true for every positive into japan. So part A in party. We say that A plus the implies that S. N. Is true or any for every positive into your end or natural number. And yeah, okay. And that's the final part of the exercise.

Yeah. Mhm. In this question here is Senate. Mhm. Two plus four. Let's eat plus two in chicken too. two n plus one minus mm. First verify. S one If I need to call to one then mm Miss one becomes two in Flesh. -2. All right. This is one plus one minus two. This is two square here. 4 -2. Well this history is going to Mhm. Mhm. Is to manage through SK is the statement. This is too plus four plus eight plus. Okay this is two. Okay. Plus 1 -1 two. No K plus one is the statement. This is 2-plus 4 plus a plus. Okay This is two K plus one. It's chicken to okay plus two minus two. A chiming s cage too. They teach two plus four plus eight plus two. Okay this is To a plus one minus. Mhm. We are put this age then we have. This is 2-plus 4 plus eight plus. Okay. Plus two K Plus one. This is two K Plus 1 -2. Then plus took a plus one. This is two into the air pollution minus two. This is too okay. Plus one plus one minus two. This is okay. Plus one, two plus four plus eight plus plus to care. And this is here plus one chicken too. Okay. Plus one minus two. Mhm. Hence if a skate through then Yes. Okay. Plus one is also true. Therefore, yes, dan. This is two plus four plus eight. This is total different plus one minus two each. True. For your nipples are training together. Mm. Uh huh. You said yes and two plus four plus eight plus two in article two To the department plus 1 -2. Is that true? Far legitimate Desert in president. Thank you.


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