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Let m be positive integer 80 that mn < " Let Gm be the subgroup of Sn composed of e Sn S0 thate{1,.- m} il i € {1,.. m}; o(j) € {m +1,.."} ...

Question

Let m be positive integer 80 that mn < " Let Gm be the subgroup of Sn composed of e Sn S0 thate{1,.- m} il i € {1,.. m}; o(j) € {m +1,.."} if j € {m +1,..n}Show that Gm Sm * Sn-m-

Let m be positive integer 80 that mn < " Let Gm be the subgroup of Sn composed of e Sn S0 that e{1,.- m} il i € {1,.. m}; o(j) € {m +1,.."} if j € {m +1,..n} Show that Gm Sm * Sn-m-



Answers

Suppose a and b are real numbers (0<b<a). Prove if n is a positive integer then a^n-b^n?na^n-1(a-b)

Well we have a question in which we need to proof one plus a place is choir one. Place a place is quite plus AQ plus areas to the power and equal to one minus areas to the power and plus one. And that's when by one minus k. Far uh not equal to one. Otherwise it will become and finite or I'm defined this is the some national geometric provisions geometric series. Okay so let us start with the proof step 11. They will be assuming let is the through four and equal to zero. Oh if we plug in an equal to zero this will be areas to the power zero equal to one minus areas without one by one minus eight. Now it is to power zero. We're playing here. No plumbing here. Okay one equal to one minus a by one minus eight. So remember we have plugged in an equal to zero here for left hand side and and equal to zero. Head for right inside one. So we can say that we can see that left hand side equal to right inside. So this is true. Now you will be assuming that let us assume that this is true for and equal to Okay, let's win. So for an equal to keep this one, we will be just writing sorry an equal to K after zero will be assuming this to be true for anyone to care. So this will be one. Place a place a square. Place a cute place. Plus areas to the power. Okay, equal to it. Is the barricade equal to one minus areas to the father. Shapeless one by one minus a. Okay, so we'll be using this expression expression number one, equation number one. We should save it all. This is it has right inside and left inside. The most important is we have to prove that we are this is true. Far and equal to K plus one. This we have to prove. So let us play in here. One plus A plus a squared plus AQ players. It is the power care plus it is the power K plus one. Equal to one minus areas to the power unit planning. And K plus one in place of K. K plus one plus one, divide by one minus E. No from equation when we have value of this expression as one minus one minus it is to the paperless one divided by one minus A and plus a building. And as it is plus, it is about a pleasant sorry, equal to one minus areas to the power K plus two by one minus a. Okay, if we just would prove that left inside it right inside so we'll be able to prove the expression Okay, let us concentrate on this. This will become one minus areas to the provider. K plus one. So we are writing LHs and here will be one mindset because we're just adding it up. Bless one minus a. Into eight days to the power Okay. Place when mhm Equal to one minus it is to the power K plus one plus areas to the power people is when minus here is to the power kate glass. When minus when? Yeah, I just opened the back it uh It is the power people. Yes, divided by one minus K. These two will get cancelled out and we will be left with one minus Oh sorry this is this indicate person because I am A has our one and this is the same, basically the same. So we have to edit the power of Okay, place to divided by one of my essay which is it will to our right hand side, it is equal to uh right inside. So we can see that that expression is being approved using mathematical induction. Thank you so very much.

Question given us when it took problematic particular election that one Q plus two cube plus three Q two and Q is equals to one plus two plus three. Dylan all square. Not approved by mathematical election. We just solid for let's say for and equals to one. And in that case our left hand side is one Q. Which is equal to the right hand side. Since that is one square and that is also one. Now when you look for any calls to K now we assume that it is true for N equals two. K. Assumed it is true and therefore we get one Q plus two cube plus kick you is equals to one plus two plus. Okay hold square. And now we need to prove for any close to K plus one. So when we write this year we get left hand side as once Q two Q bless KQ plus K plus one. Thank you. And our left hand side. Sorry, 19 side is one plus two plus K plus k plus one whole square. Now let's start with the left hand side. So here this entire term can be replaced by this value which is one plus two. Bless less clear square. And we lived with K plus one. Okay. So now we have two dumps and we need to simplifies us that we get this. If both are equal now to do that we can just neglect this step. And what we can use is summation formula. So here this is one plus two plus three T. K plus one cube. And this is some of Uh huh. Where are cube? Sorry? Where are they starting from? One till it's a K plus one. And we know some mention of our cube is if they are going to keep this one. So this escape plus one square. Okay. Plus one. And here we have K plus one plus one since N plus one whole square by four. And on simplifying get K plus one square. Okay. Plus two. Great. 44 So this is our left hand side. Now let's solve the right hand side part. So here we can apply submission formula inside the square down. So this is one plus two plus three plus K. K plus one. So this can be written as some of our our starting from one tree. K plus one whole square summation of our is given by formula N. But here an escape plus one. K plus one. Okay. Plus one plus one by two whole square. And this gives us K plus one square. This becomes K plus two squared over. And that is the same as this and total by induction. Thank you.

Okay. So I need to prove that A. Is greater than zero and there exists and such that one over less than a. This and then so we're going to use the property where A. Is greater than zero and B. Is greater than zero. Then for some integer N. We have that in times A. This period can be yes, it's a comedian happen. So first we'll say let's let A equal one and be equals A. Then there this property must exist and then such that and one times one. Which is it was a be greater than a. Okay in other words this this um and one description A. So now and say well let's let a little A. And this baby is going to equal on it. Okay, so by the same property there must be some other some different yeah such that A. It's greater than one over and okay. And this comes from did kind of skip a step. Um that's derived from if we'd written it out as and to a. Being greater than one the bible side. Yeah to it this thing so now we're just gonna let N. N. So the N. B. The maximum of these two introduce then what we have is that end it's greater than or equal to N. One and and it's greater than or equal to and to the only what happens to be more such ad A. Such that which that A. Is less than and one is less than or equal to then. Okay that's going to follow up the first one and yeah one over N. It's just an equal to one over and two less than A. That follows from that second round. We did okay so we just got to combine these two pieces, plug them together we get that A. Is greater than one over N. But less than and Okay and so and that is our solution.

Zero is less than B is less than a. Let's consider the function A four X. Is equal to expire in and is it and teacher great and equal to do now there is a theory called mean value to you. This function is a polynomial. It's continuous and differentiable. In an interval beat way. It's continuous enclosing to Albert way and differential. An opening to a gateway. Right? Very function is continuous in a closed into a gateway and differential and opening to a gateway then like green just mean value to them like ranges mentality and says that F a B minus ffb divided by a minus B is equal to a dash of something between be any. A very powerful freedom in calculus. Lagrange is humanitarian. Let's use this. So what is my fook? Fook will be a parent minus B. Baron divided by a minus B. Is equal to a dash of. See what is the derivative of this function? It is an ex power in minus one. This is the radio off function. Experiment by power. So what will be a flash of C. N. C. Bar and minus one? All right now you can see here sees between B. And so that means he's less than me. You see is less than a C. Bar and minus one is less than a bar and minus one. Multiplying and pull tight N. C. Bar and minus one is less than in a bar in minus one. So this quantity is less than in a par and minus one. So that would imply a par n minus B part and divided by a minus B is less than any power and minus one for all and greater than or equal to two for one. What will happen is it will be exactly equal because a pas one minus people. One divided by a minus B. It's compared with one paper, one minus one. This is one and 180 is one. So for one when it is one, this particular inequality is actually an equal. But when you increase one, that is when you take higher than 12345 all natural numbers starting with two. This inequality will hold two and we included by Lagrange is mean value.


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