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Author Topic:   Geometric mean

samalbee posted 04-19-122 10:12 ET (US)
Geometric mean
The geometric mean g of a set of positive values x1, x2 ... xn is equal to the nth root of the product of these values:
g=n√(x1⋅x2⋅...⋅xn)
The geometric mean g of two positive numbers a and b is thus the - [url=https://domyhomework.club/geometry-homework/]geometry homework help[/url]  (square) root of their product:
g=√(a⋅b)
Examples:
The geometric mean of the numbers 4 and 9 is 6. In a right triangle - [url=https://domyhomework.club/]do my homework for me[/url] (according to the height theorem), the height h belonging to the hypotenuse is the geometric mean of the hypotenuse sections p and q, i.e., it holds:
h=√(p⋅q)
The geometric mean of two positive real numbers a and b is always smaller than their arithmetic mean, i.e., it holds:
√(a⋅b) < (a+b)/2
(a,b∈R;a, b>0).
This can be easily shown: From the above inequality it follows via
2√(a⋅b) < a+b
4ab<(a+b)^2 = a^2+2ab+b^2
0<a^2-2ab+b^2
with 0<(a-b)^2 a true statement.
Harmonic mean
The harmonic mean h of a set of positive values - [url=https://domyhomework.club/html-assignment-help/]html homework help[/url] - x1, x2 ... xn is equal to their number n divided by the sum of the reciprocals of these values:
h=n/(1/x1+1/x2+...+1/xn)
The harmonic mean h of two positive numbers a and b is therefore:
h=2/(1/a+1/b)=2ab/(a+b)
Example:
The harmonic mean of the numbers 4 and 9 is 72/(4+ 9)=72/13≈5.54.
For the arithmetic mean x¯, the geometric mean g and the harmonic mean of positive real numbers, the general rule is:
h<g<x¯ or (in the case of two positive real numbers a and b) specifically:
2ab/(a+b)<√(a⋅b)<(a+b)/2
(The correctness can be easily confirmed by recalculation).
Example:
For the three numbers 5, 8 and 11 is
h=3/(1/5+1/8+1/11)≈7,21;
g=3√(5⋅8⋅11)≈7,61;
x¯=(5+8+11)/3=8
and thus the relation h<g<x¯ holds.
Read also:
[url=http://kwtechjobs.com/?author=215725]Development of time measurement[/url]
[url=https://steelpantherrocks.com/community/fanthers/users/70223]Regular polygons[/url]
[url=https://vardell.biz/members/samalbee/profile/]Early Modern Times – an overview[/url]
[url=https://mindvalley.kl.tis.edu.my/user/sam_albee]Vasco da Gama's first voyage to India 1497-1499: From the ship's journal[/url]
odobasi25g posted 06-08-122 03:45 ET (US)
Hi Samalbee,
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gthert posted 06-08-122 03:54 ET (US)
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Author	Topic: Geometric mean
samalbee	posted 04-19-122 10:12 ET (US) Geometric mean The geometric mean g of a set of positive values x1, x2 ... xn is equal to the nth root of the product of these values: g=n√(x1⋅x2⋅...⋅xn) The geometric mean g of two positive numbers a and b is thus the - [url=https://domyhomework.club/geometry-homework/]geometry homework help[/url] (square) root of their product: g=√(a⋅b) Examples: The geometric mean of the numbers 4 and 9 is 6. In a right triangle - [url=https://domyhomework.club/]do my homework for me[/url] (according to the height theorem), the height h belonging to the hypotenuse is the geometric mean of the hypotenuse sections p and q, i.e., it holds: h=√(p⋅q) The geometric mean of two positive real numbers a and b is always smaller than their arithmetic mean, i.e., it holds: √(a⋅b) < (a+b)/2 (a,b∈R;a, b>0). This can be easily shown: From the above inequality it follows via 2√(a⋅b) < a+b 4ab<(a+b)^2 = a^2+2ab+b^2 0<a^2-2ab+b^2 with 0<(a-b)^2 a true statement. Harmonic mean The harmonic mean h of a set of positive values - [url=https://domyhomework.club/html-assignment-help/]html homework help[/url] - x1, x2 ... xn is equal to their number n divided by the sum of the reciprocals of these values: h=n/(1/x1+1/x2+...+1/xn) The harmonic mean h of two positive numbers a and b is therefore: h=2/(1/a+1/b)=2ab/(a+b) Example: The harmonic mean of the numbers 4 and 9 is 72/(4+ 9)=72/13≈5.54. For the arithmetic mean x¯, the geometric mean g and the harmonic mean of positive real numbers, the general rule is: h<g<x¯ or (in the case of two positive real numbers a and b) specifically: 2ab/(a+b)<√(a⋅b)<(a+b)/2 (The correctness can be easily confirmed by recalculation). Example: For the three numbers 5, 8 and 11 is h=3/(1/5+1/8+1/11)≈7,21; g=3√(5⋅8⋅11)≈7,61; x¯=(5+8+11)/3=8 and thus the relation h<g<x¯ holds. Read also: [url=http://kwtechjobs.com/?author=215725]Development of time measurement[/url] [url=https://steelpantherrocks.com/community/fanthers/users/70223]Regular polygons[/url] [url=https://vardell.biz/members/samalbee/profile/]Early Modern Times – an overview[/url] [url=https://mindvalley.kl.tis.edu.my/user/sam_albee]Vasco da Gama's first voyage to India 1497-1499: From the ship's journal[/url]
odobasi25g	posted 06-08-122 03:45 ET (US) Hi Samalbee, Thank you very much because what you share in this article is extremely helpful for me and many others. <a href="https://driftboss.io">drift boss</a>
gthert	posted 06-08-122 03:54 ET (US) I'm overjoyed because this is great <a href="https://driftboss.io">drift boss</a>. Always try your best.