Geometric mean Definition and 19 Threads

Given are a fixed point ##P## and a fixed circle ##c## with the radius ##r##. Point ##P## can be anywhere inside or outside the circle. I now draw two arbitrary lines ##l_1## and ##l_2## through the point ##P## in such a way, that both lines intersect with the circle ##c## in two distinct...

Homework Statement We have a normal 6 sided dice marked from 1 to 6. There is an equal chance to get each number at every roll. Let's put 1&2 as A type, 3&4 as B type and 5&6 as C type. We roll the dice over and over until we get a number of every type. Let X be the number of rolls. We are...

The Beer-Lambert law gives the intensity of monochromatic light as a function of depth ##z## in the form of an exponential attenuation: $$I(z)=I_{0}e^{-\gamma z},$$ where ##\gamma## is the wavelength-dependent attenuation coefficient. However, if two different wavelengths are present...

Homework Statement Let ##\{a_n\}## be a sequence of positive numbers such that ##\lim_{n\to\infty} a_n = L##. Prove that $$\lim_{n\to\infty}(a_1\cdots a_n)^{1/n} = L$$ Homework EquationsThe Attempt at a Solution Let ##\epsilon > 0##. There exists ##N\in\mathbb{N}## such that if ##n\ge N## then...

Homework Statement The problem is stated as follows: "The result in Problem 1-7 has an important generalization: If ##a_1,...,a_n≥0##, then the "arithmetic mean" ##A_n=\frac {a_1+...+a_n} {n}## and "geometric mean" ##G_n=\sqrt[n] {a_1...a_n}## Satisfy ##G_n≤A_n## Suppose that ##a_1\lt A_n##...

Homework Statement Hello. There is a financial metric called time weighted rate of return, which is computed using the following formula: 1) if we compute daily returns, or other returns within a year: r tw = (1+r1) x (1+r2) x...x (1+r nth year), where r tw is the time weighted rate of return...

Homework Statement Hello! I am trying to compute the bi-geometrical mean on data that contains negatives. But before that I wanted the test the formula that accounts only for positive values using the sum of their logarithms. By doing so I don't get the result I compute by using the "usual" geo...

In a geometric mean equation, say 2 x 8 = 16, or a x b = c, what are the words we would use to describe the numbers or terms? Specifically, if you know 'a' and 'c', what do you call 'b'? For example, in a normal multiplication, a x b = c, 'a' is the multiplicand, 'b' is the multiplier, and 'c'...

Homework Statement Let ## a,b,c \in \mathbb{R}^{+} ##. Prove that $$ \sqrt[3]{abc} \leq \frac{a+b+c}{3}. $$ Note: ## a,b,c ## can be expressed as ## a = r^3, b = s^3, c = t^3 ## for ## r,s,t > 0##. Homework Equations ## P(a,b,c): a,b,c \in \mathbb{R}^{+} ## ## Q(a,b,c): \sqrt[3]{abc} \leq...

Homework Statement GMR_{hollow cylinder}=Re^{-Kμ} where K=\frac{AR^4-R^2r^2+Br^4+r^4ln(R/r)}{(R^2-r^2)^2}, where R is the outer radius and r is the inner radius, and mu is the relative permeability. We are to determine the numerical values of A and B. I am stumped on how to begin attempting...

Why is the geometric mean used to define the center frequency of a bandpass filter instead of the arithmetic mean? I read in this book that 1. All the lowpass elements yield LC pairs that resonate at ω = 1. 2. Any point of the lowpass response is transformed into a pair of points of the...

$a,b,c$ are any three positive numbers such that $a+b+c=1$. Prove that $$ab^2c^3 \leq \frac{1}{432}$$

Hi, I have the following equation: \gamma=\frac{1}{\frac{1}{N}\sum_{n=1}^N|\lambda_n|^{-2}} where lambdas are the eigenvalues of an N-by-N circulant matrix A. I used two properties to bound the above equation...

Homework Statement Let r_{1}, r_{2}, ... , r_{n} be strictly positive numbers. Show that the inequality (1+R_{G})^{n} \leq V is true. Where R_{G} = (r_{1}r_{2}...r_{n})^{1/n} and V= \Pi_{k=1}^{n} (1+r_{k}) Homework Equations The Attempt at a Solution I've...

Homework Statement let r_{1}, r_{2}, ... , r_{n} be strictly positive numbers. Suppose an investment of one dollar at the beginning of the year k grows to 1+r_{k} at the end of year k (so that r_{k} is the "return on investment" in year k). Then the value of an investment of one dollar at...

I have the following mapping (generalized geometric mean): y(i)=exp\left[{\sum_j p(j|i)\log x(j)}\right]\\ ,\ i,j=1..N where p(j|i) is a normalized conditional probability. my question is - is this a contraction mapping? in other words, does the following equation have a unique...

Hey, (sin A + sin B + sin C)/3 >= \sqrt[3]{}(sin A*sin B*sin C) I know this is true by Arithmetic mean always greater than geometric mean... but is there any other way of proving this?

I don't know why I can't figure this one out tonight. I just can't think straight and I am hoping someone can help ASAP. Here is the question: In 1998 revenue from gambling was $651 million. In 2001 the revenue increased to $2.4 billion. What is the geometric mean annual increase for the period?

I have a question that I would like your assistance to see if I have the correct info: In 1990 there were 9.19 million cable TV subscribers. By 2000 the number of subscribers increased to 54.87 million. What is the geometric mean annual increase for the period ? Answer...