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Show that the equationi.i! = (n + 1)! - 1 1=1Holds and is odd...

Question

Show that the equationi.i! = (n + 1)! - 1 1=1Holds and is odd

Show that the equation i.i! = (n + 1)! - 1 1=1 Holds and is odd



Answers

Prove that if $n$ is an integer, then $\lfloor n / 2\rfloor= n / 2$ if $n$ is even and $(n-1) / 2$ if $n$ is odd.

In discussion. We have to prove that a function off following form is hard. I love eggs. We could do to and plus one X to the power to time and plus one plus two time and minus one X to the power two time on minus one Close. So close. Okay, Acts to the board, plus a one next when Ah, off a minus eggs. So we get eggs too. And plus one minus x to the power two and bless one bloods a two on minus one minus x to the power two and minus one plus so on X two people Extremely a tree extra. The power three minus X to the power three plus everyone minus eggs. Now you can see that in this portion. Here we get negative value. You also get negative value. And in this you know that best Sam. So it'll be a minus x to the power to end. So we get father to value but minus X to the power one. So we get negative. So we multiply positive and negative. Then we get negative, so it will be negative. And now you can see that in this minus X to the power to end. So it means we will get positive value here. But you can see that minus X to the power minus one. So here we get negative value so you can take common not going to ST in this because all of drums are negative, so we can take minus. It is a greatest common factor. So a two and plus one x to the power to and plus one plus a two and minus one x to the power to end minus one plus and so on, plus a tree X to the Power cube less it one x So we know that it is epics, so you can write like that F minus eggs equals toe minus eggs. Therefore, uh, off eggs it is that is your final answer.

In discussion. We have to prove that a function off following form is hard. I love eggs. We could do to and plus one X to the power to time and plus one plus two time and minus one X to the power two time on minus one Close. So close. Okay, Acts to the board, plus a one next when Ah, off a minus eggs. So we get eggs too. And plus one minus x to the power two and bless one bloods a two on minus one minus x to the power two and minus one plus so on X two people Extremely a tree extra. The power three minus X to the power three plus everyone minus eggs. Now you can see that in this portion. Here we get negative value. You also get negative value. And in this you know that best Sam. So it'll be a minus x to the power to end. So we get father to value but minus X to the power one. So we get negative. So we multiply positive and negative. Then we get negative, so it will be negative. And now you can see that in this minus X to the power to end. So it means we will get positive value here. But you can see that minus X to the power minus one. So here we get negative value so you can take common not going to ST in this because all of drums are negative, so we can take minus. It is a greatest common factor. So a two and plus one x to the power to and plus one plus a two and minus one x to the power to end minus one plus and so on, plus a tree X to the Power cube less it one x So we know that it is epics, so you can write like that F minus eggs equals toe minus eggs. Therefore, uh, off eggs it is that is your final answer.

So we want to show the ceiling of and over two is n plus one Over two went on his side that NBR by definition and see culture two. Okay. Last one for some K. In the integers. So notes. No dad. Okay. Is equal to And -1/2. I just played with this cake. A base up. We don't have The ceiling for # two. We have an art. So to K plus one. You were too which is equal to um to carry over two plus one half. Okay. We can separate this left with gave plus 1/2 just equal to. Okay. Plus. Uh huh. Okay. Is an insider. So it's stealing is just K. The ceiling of 1/2 is one. Remember we said uh to note case and minus 1/2. So you can substitute that back in and minus 1/2. Plus. Why? This gives us what and minus one last two over to which is equal to. And plus one. That's it.

Okay. In this question, we want to prove that end square is equal to is equivalent to one more eight. When n is an odd positive integer So we basically want to Seoul and square and squid mud. Thanks. So this is equivalent off solving and mud aids squared and he wants it more than eight. So now it's useful to think about what values this first more can take. So if n isn't odd, didja zero copy this value But it could be you could take on one because you can't take onto it can take out three can't taken four We'll take a five on dhe seven. So then, from one you have one squared more. Eight. So this is just one. You have three squid which gives you night mud eight. So this is 15 squared gives you 25 which is 20 by minus What equivalent to minus y three times by eight 18 So this number right here it has freed give you 24 So 25 minus 24 which is just one finally seven gives you seven squared gives you 49 so this is equivalent to 49 months by 56 times by eight 18 So it's six times 8 48 now 49 minus 48 which is simply one. So therefore, n squared is equal to one month. Eight if m is a positive integer.


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