hardest topic (1 Viewer)

[wogboy]

Actually working backwards can be quite helpful for you to solve the question. I end up working backwards quite alot with these inequations

But the AM/GM stuff is still quite important though for some questions like this one.

 

[Jason]

Hang on! We haven't done any of this AM/GM stuff - is it a part of the course? Could someone please explain to me what it is and how to do it?

 

[wogboy]

Yeah you should know that arithmetic mean of two different positive numbers is ALWAYS greater than their geometric mean. What does this mean?

Arithmetic mean of a & b (AM) = (a + b) / 2
Geometric mean of a & b (GM) = sqrt(ab)

AM > GM for all positive a and b. (acutally AM = GM if and only if a=b)

The proof for this is:

[sqrt(a) - sqrt(b)]^2 >= 0 (all squares of real nubers are
positive obviously)
a - 2sqrt(ab) + b >= 0

a + b >= 2sqrt(ab)

therefore,

(a+b)/2 >= sqrt(ab)

equality only holds if a=b

For more than two numbers (e.g. a, b, and c) look at McLake's method of working it out.

This isn't only useful in itself, but you can apply this rule to other harder inequality questions to solve them.

 

[Dumbarse]

madd, thanx!! we were never taught that

 

[Dan_the_man]

I don't think we ever learned that either
teacher

 

[Raser]

what's the basic formula thingy for HM??

(1/a+1/b)^-1
something like that???
or are there's squares around the a and b s

 

[Raser]

hey i found it
HM:

n
-----------------------------------------
(1/a1 + 1/a2 + 1/a3 + ..... 1/an)

the numbers after the a's are subscripts just incase you don't see it

proof is just but letting the a's in the AM and GM = 1/a
and you should be able to see it

 

[school-spew]

some of the past paper mechanics questions are killers.
is anyone else freaking out yet about the prospect of actually sitting for this damn paper?

 

[Jason]

Thanks for the AM/GM thing - now I'll have a small chance at the harder 3U questions!

 

[Dumbarse]

what does HM stand for??

 

[Raser]

harmonic means
it's AM>=GM>=HM

 

[McLake]

Originally posted by Raser
harmonic means
it's AM<=GM<=HM

and HM = ???

 

[Dumbarse]

yes what does HM = ??

 

[Raser]

n
---------------------------------------
(1/a1 + 1/a2 + 1/a3 + ..... 1/an)

the numbers after the a's are subscripts just incase you don't see it

proof is just but letting the a's in the AM and GM = 1/a
and you should be able to see it

i type dit once already

in simplified form i guess it's

2/(1/a + 1/b)

 

[spice girl]

A 'power mean' P(k) is this:

P(k) = ( (a1^k + a2^k + a3^k + ... + an^k) / n )^(1/k)

Ao AM - arithmetic mean is when k = 1, HM - harmonic mean is when k = -1, for those interested, QM - quadratic mean is when k = 2, cubic mean is when k = 3, and geometric mean is when k approaches 0.

A general rule is if a1, a2, ..., an are all positive, and if x > y, then P(x) >= P(y). (equality a1=a2=...=an)

Thus QM >= AM >= GM >= HM

 

[Dumbarse]

without putting it in this shatty form

n
---------------------------------------
(1/a1 + 1/a2 + 1/a3 + ..... 1/an)

which i have no idea what the dotted line means,
or the a1 a2 a3 bizzo
what is this subscript crap,

can someone just tell me the formula for HM!?!

 

[Raser]

n /(1/a1 + 1/a2 + 1/a3 + ..... 1/an)
happy

my god
i should have kept my mouth shut

 

[McLake]

Originally posted by Raser

my god
i should have kept my mouth shut

Y?

 

[Raser]

coz by bother saying something that won't be understood

 

[Dumbarse]

i didnt mean to blow up,

i still dont get, what are the a1 a2 a3

whats an example of a question??