[/quote]
,
does the sequence 
defined by the initial condition 
and the recursion 
have 
for all 
Is there any reason why we shouldn't write 
instead?
be real numbers such that 
and 
and determine the equality cases.
be positive real integer such that 
Prove that
such that 
and 
is defined by the conditions ![\[ x_0 = a, x_1 = b, x_{n+1} = \frac{x_{n - 1} + (2n - 1) ~x_n}{2n}\]](http://latex.artofproblemsolving.com/9/5/3/953123f35abcc7490ff1fedc76b356bdf161759a.png)
for 
where 
are given numbers. Express 
concisely in terms of 
satisfying :![\[n^{2p}=m^2+n^2+p+1\]](http://latex.artofproblemsolving.com/7/4/e/74ee592cb8b3f3a3996ea34f40b78fde9d3672fc.png)
and 
are integers and 
is a prime number.
with 
coprime of the open interval 
is covered by the closed interval ![$\left[\frac{p}{q}-\frac{1}{4q^{2}}, \frac{p}{q}+\frac{1}{4q^{2}}\right]$](http://latex.artofproblemsolving.com/6/0/3/60330f376a5ce44ff7c9584247b7a213f5463791.png)
.
is not covered by any of the above closed intervals.
Approach: I'm trying to factor 
to get 
.
such that 
divides 
.
![$a,b,c \in [-\frac{1}{2}, \frac{1}{2}]$](http://latex.artofproblemsolving.com/5/0/b/50bc691baa4e4bbef868e4799929c8c93d1b6f1a.png)
.
Let ![$a,b,c \in [-2,2]$](http://latex.artofproblemsolving.com/8/0/6/8066d4353091772c401e3e34f5499a8fa2bd9991.png)
.
be positive real numbers such that 
Prove that 
.
positive integer quadruples satisfy:
![\[\sum_{k=1}^n\frac{\phi(k)}k\ln\frac{2^k}{2^k-1}\ge 1-\frac 1{2^n}.\]](http://latex.artofproblemsolving.com/3/d/3/3d32207efff8052214e65aca2505c9e4e7bb581c.png)
Y by Adventure10
Need to find elements of order
.
14 reflections (of order 2) and 14 rotations.
First look at
.
1 (mod 7)A unique Sylow 7-subgroup of order 7. No reflections in this subgroup (as they are of order 2).
There are 7 rotations (including identity).
So, if <x> are rotations and <y> are reflections, then in the Sylow 7-subgroup of order 7 there are only elements generated by x.
are of order 2. 
is of order 7? No elements of order 4 in in the Sylow 7-subgroup.Looking at
.
1 (mod 2)The Sylow 2-subgroup is of order 4.
as we have
, does this mean that there are no elements of order 2 in the Sylow-2 subgroup, but only elements of order 4.I need to find:
(1) elements of order
in Dih(28)(2) list the Sylow 2-subgroups and the Sylow 7-subgroups.
Not sure if I am going in the right direction with this...
Any help would be appreciated!
.
)
.
has a total of 
permutations. It is describes the symmetry of a regular 
clockwise, we have 
.
there are 
permutations of order 
.
