inductive proofs

[shakalandro]

I have this problem where I need to prove inductively the problem attached for positive integers n and positive real numbers x[sub:148855tf]i[/sub:148855tf].

The problem also says that "It does not seem to be possible to give a direct proof of this result using induction on n, however it can be proved for n = 2[sup:148855tf]m[/sup:148855tf] for m greater than or equal to 0 by induction o m. The general result now follows by proving the converse of the usual inductive step: if the result holds for n = k +1, where k is a positive integer, then it hold for n = k."


I'm not even sure what the problem looks like once n is substituted out for 2[sup:148855tf]m[/sup:148855tf]. And what does x[sub:148855tf]i[/sub:148855tf] even mean in a product or summation?

 

[Deleted member 4993]

I have this problem where I need to prove inductively the problem attached for positive integers n and positive real numbers x[sub:203gklpx]i[/sub:203gklpx].

The problem also says that "It does not seem to be possible to give a direct proof of this result using induction on n, however it can be proved for n = 2[sup:203gklpx]m[/sup:203gklpx] for m greater than or equal to 0 by induction o m. The general result now follows by proving the converse of the usual inductive step: if the result holds for n = k +1, where k is a positive integer, then it hold for n = k."


I'm not even sure what the problem looks like once n is substituted out for 2[sup:203gklpx]m[/sup:203gklpx]. And what does x[sub:203gklpx]i[/sub:203gklpx] even mean in a product or summation?

Your *.tiff does not display on my screen.

 

[stapel]

For those unable to view the image, the text it contains is as follows:

. . . . .(1/n)sum[i=1,n] x[sub:3cuqwiuo]i[/sub:3cuqwiuo] > (prod[i=1,n] x[sub:3cuqwiuo]i[/sub:3cuqwiuo])[sup:3cuqwiuo]1/n[/sup:3cuqwiuo]

This may be formatted (rather than simply typed) as:

. . . . .\(\displaystyle \frac{1}{n}\, \sum_{i=1}^n \, x_i\, \geq \left( \prod_{i=1}^n\, x_i \right)^{\frac{1}{n}\)

 

[Deleted member 4993]

So your problem is asking you to show that the "algebraic mean" is greater than or equal to the "geometric mean".

Do a google search with those terms and show us what do you find.

 

[shakalandro]

I found the relationship that the geometric mean is equivalent to the sum of the natural logs of each x, all exponentialized to e and then multiplied by 1/n

http://en.wikipedia.org/wiki/Geometric_mean

tricky, but it makes sense