I am sorry for previous misunderstanding. Here's the correct one?

Question

If r<1 and positive , and m is a positive integer ,show that
{(2m + 1)r^m}(1-r) < 1-r^2m+1???

O.K. may be it is better to put it this way
the R.H.S. is 1 - (r)^2m+1

Answer 1

Hi integrity,
make the inequality look like........
(1- r^2m+1) / (1-r) > (2m+1) r^m ;
LHS is nothing but.....1+r+r^2+.........+r^2m ;
{use G.P. to get back that}
the inequality reduces to..........
(1+r+r^2+.......+r^m-1) + (r^m+1 +...........+r^2m) > 2m*r^m
{r^m gets cancelled both sides}
using A.M > G.M........
Observe there r 2m terms in lhs......
So, A.M= {(1+r+.....+r^m-1) + (r^m+1+...........+r^2m)} / 2m
G.M=(1*r*...*r^m-1 *r^m+1 *........*r^2m) ^ 1/2m
=(r ^ [1+2+....2m - m])^1/2m
=(r ^ [2*m^2]/2m)
=r ^ m ;
Therefore, lhs> rhs {by taking 2m to other side in AM>GM}
Hence, proved.

NOTE:here,AM =/= GM .equality holds when all terms r equal , as r=/=1 {given r<1} ;AM>GM.

Answer 2

Just answer this one question: Where is the positive evidence for ancient astronauts? Lets face the truth: By the way how ancient astronaut "theories" work, we could also claim that the Cologne Cathedral was build 1000 years ago as rocket launch complex. Without looking at the cultural context, that building makes no sense, and if you then also add a few lies to the story (The Cologne Cathedral was actually finished in the 19th century) it sounds more impressive and spectacular enough to get media attention. Ancient astronaut "theories" are just fiction, and no scientific theory. It is not based on proper evidence or evidence that is treated out of context. Like von Däniken simply turning ancient reliefs upside down and claiming they are something alien, ignoring the context. PS: About the 30 ton stone. Why is that so incredible to you, that "100" humans can move a 30 ton stone (and that even upwards a slope), if you can alone already push a 1.5 ton car forward on a level surface. Yes, the car has less friction. Granted. But it does move. Given enough thrust, even the Pentagon could fly. If you have enough humans, you can move the stone. There should be no question about that. Now why "100"? Because the Egyptians say they used ~120 humans for moving a 17 ton statue - there is such a drawing on the wall of a tomb. You must not believe the drawing, it could be fiction. Right. But it is credible when testing the physics. You would need 240 humans for moving a 30 ton stone at the same speed as 20 humans can pull a 2.4 ton block. That might be almost unbelievable, but on the other hand, can you imagine that a human can run 100m in less than 10 seconds? I would say, in an optimistic estimate, you and all your friends would be limited to 13-15 seconds for the distance. Yet, I think you have no doubts that some humans can do that. maybe all your friends could not build a clipper ship. Is that now impossible? and if I would put all your friends on such a ship, you would likely be unable to control it. That doesn't mean that it is impossible for all humans, it only means you lack the skills or the experience to tell what could be possible. You would need to try it, look at how you tried, maybe improve it. And why is "descending from heaven" a adequate proof of air or space travel for you? At the time when such mythology was created, Earth was flat, Sky a spherical shell, and the sun in many cultures a god that rode over the sky, instead of flying, with the sun descending below Earth or the Oceans. Cultural context. And yes, they used tree stems - nearly all wood that was imported from Lebanon during the construction must have been used for that. But they did not use rolling tree stems, they had then fixed !!!into!!! the ramp, and lubricated them with water (from the nearby Nile). That has less wear of the tree stems, and almost as little friction, when using enough water.

Answer 3

You are still missing parentheses on the right hand side.

Answer 4

u see this thing is a little tricky

Answer 5

Well,you're 1 great genius...perhaphs I am stupid???but i dunno.