Geometric Definition and 813 Threads

Let x1, ..., x25 be such positive integers that x1⋅x2⋅ ... ⋅x25 = x1 + x2 + ... + x25. What is the maximum possible value of the largest of numbers x1, x2, ..., x25?

Solution to the problem tells us that ##S_5 + i S_6## is the sum of the terms of a geometric sequence and thus the solutions should be : $$S_5 = \frac{\sin( (n+1) x)}{\cos^n(x) \sin(x)},\,\,\,\, S_6 = \frac{\cos^{n+1}(x) - \cos((n+1)x)}{\cos^n(x) \sin(x)} , x \notin \frac{\pi}{2} \mathbb{Z}$$...

I'm using the sum of a geometric series formula, but I'm not sure how to find the ratio, r. The n is confusing me. The solution is below, but I'm having trouble with the penultimate step.

The matrix ##A## in question is ##\dfrac{1}{3} \left(\begin{array}{rrr} -2 & 1 & -2 \\ -2 & -2 & 1 \\ 1 & -2 & -2 \end{array}\right)## One can easily verify that ##AA^t=I##, hence an isometry. To find its geometric meaning, one can proceed to find ##U=\text{ker} \ (F-I)=\text{ker} \...

My question is Why is the sum to infinity used as opposed to Sum to n? and How can I deduce that the sum to infinity must be used from the question?Total Distance = h + 2*Sum of Geometric progression (to infinity) h + 2*h/3 / 1-1/3 h + 2h/3 *3/2 = h + h = 2h At first I did sum to infinity...

There is a property to geometric distribution, $$\text{Geometric distribution } Pr(x=n+k|x>n)=P(k)$$. I understand it in such a way: ##X## is independent, that's to say after there are ##(n+k-1)## successive failures, ##k## additional trials performed afterward won't be impacted, so these ##k##...

Let the (multi-vector valued) “inner product” between a j-vector U and a k-vector B be defined as the (k-j) grade part of the geometric product UB, (a.k.a. “left contraction”) that is, $$U\cdot B := <UB>_{k-j}$$ (0 when j > k) as is done in Alan Macdonald’s book “Linear and Geometric Algebra.”...

I was watching a TV show about these telescopes being transported up to a high plateau in Chile. Looking at the Wiki page, it seems that they are to be arranged in random geometry, although maybe there is some pattern to the arrangement. Does being in some pattern - or being in a random...

So, I know it can be proven using calculus, but I need the geometric one. So, I got that ^c=^d and therefor, the amount of increment in one of a, is equal to the other(^e=^b). (Also 0<a+b<Pi/2) And AP'=BP'=BD/sin(a) and BP=BD/sin(a+b) and AP=BD/sin(a-b). AP'+BP'=2AP'=2BD/sin(a) and...

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with an aspect of the proof of Proposition 2.3.8 ... Proposition 2.3.8 and its proof read as follows: In the above proof by...

Why do snowflakes freeze into complex geometric patterns?

With respect to operations, I understand why an integral is multiplied by -1 when its limits reversed. But integral is geometrically an area so reversing the limits would not be able to change neither how large is the area nor the shape of the area. Would you please explain changing the limits...

A0 = 1 A1 = 3 3(An-1) / 4(An-2) = An

If a,b, c, are in G.P and $\log_ba, \log_cb,\log_ac$ are in A.P. I want to find the common difference of A.P. Answer: After doing some computations, I stuck here. $\frac{2(\log a+\log r)}{\log a+2\log r}=\frac{2(\log a)^2+3\log r\log a +2(\log r)^2}{(\log a)^2+\log r\log a}$ How to proceed...

Greg Bernhardt submitted a new blog post The Sum of Geometric Series from Probability Theory Continue reading the Original Blog Post.

Hi, I am trying to learn Geometric Algebra by going through the book "New Foundations for Classical Mechanics" by David Hestenes. I was reading the part about reduction formula (shown below) but couldn't get the result the shown in the book. Can someone show me how iterating (1.15) gives the...

Good day everybody, I'm currently working on the Grover algorithm. You can also illustrate this process geometrically and that's exactly what I have a question for. In my literary literature one obtains a uniform superposition by applying the Hadamard transformation to N-qubits. So far that's...

Hey all I previously asked about some math structure fulfilling some requirements and didn't get much out of it ( Graph or lattice topology discretization ). It was a vague question, granted. Anyway, I seem to have stumbled upon something interesting called geometric group theory. It looks...

(1) Is there a pi in ellipse entity?Why not or yes? (2) Is there a pi in polygons entities (e.g square)? not or yes? (3) If there is pi in some geometries and other not - What is the reason to that? (4) How cloud I know that are no hidden formula of pi in a square figure that the expression in...

I am reading the book: "Vector Calculus, Linear Algebra and Differential Forms" (Fourth Edition) by John H Hubbard and Barbara Burke Hubbard. I am currently focused on Chapter 6: Forms and Vector Calculus ... I need some help in order to understand some notes by H&H following Figure 6.1.6 ...

Homework Statement A 2.0-cm-tall candle flame is 2.0 m from a wall. You happen to have a lens with a focal length of 32 cm. How many places can you put the lens to form a well-focused image of the candle flame on the wall? For each location, what are the height and orientation of the image...

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...

How is the "geometric optics approximation" exactly defined? Given all the source of visible radiation's parameters, all the apparatus, instruments, screen, etc, specifications, how can I know if, e. g. there will be diffraction, interference or other wave properties or if I'll be able to...

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...

I was reading a different post, https://www.physicsforums.com/threads/differntiating-a-circle.279719/#post-2003287 and was doing great... guys, thanks so much for this forum! HallsofIvy posted "1: You titled this "differentiation of a circle" which makes no sense. You cannot differentiate a...

Hi all, I am trying to teach myself GD&T from a textbook (Fundamentals of Geometric Dimensioning and Tolerancing - Krulikowski) and would like some feedback on a drawing I've created, please see attached. Any help/criticism would be greatly appreciated. Regards, Doc

Homework Statement Let's call the axis of the ##z## complex plain ##x## and ##y##, so a general point can be written as ##z=x+iy##. Reflect the points of the complex plain so that the mirror line of the transformation is a line parallel to the vector ##v## and it passes trough the point ##u##...

I will state the problem below. I don't quite understand what I am needing to show. Could someone point me in the right direction? I would greatly appreciate it. Problem: Let p be a natural number greater than 1, and x a real number, 0<x<1. Show that there is a sequence $(a_n)$ of integers...

Homework Statement Determine the triangles where the sides are consecutive elements of a geometric sequence and the angles are consecutive elements of an arithmetic sequence. Homework Equations The Attempt at a Solution I don't really know how to approach this problem, what the solution would...

What does 1/-1 (one divided by minus one) mean? What does -1 X -1 (minus one multiplied by minus one) mean? What are the best graphic representations of multiplication and division?

Hi, outside the mathematical proof that shows that sines of different frequency are orthogonal... is there geometric interpretation/picture of this phenomena?

Homework Statement The first three terms of a GP are X,X+2,X+3. The value of X and the fifth term is.[/B] (a)-4,1/4 (b)4,1/4 (c)2,1/4 (d)-2,-1/4 Homework EquationsThe Attempt at a Solution (x+2/x)=(x+3)/(x+2) (x+2)2=x2+3x x2+2x+4=x2+3x x=4 so i think r=(x+2)/x putting x=4 r=3/2 also...

Mentor note: Thread moved from a different forum section, so missing the homework template. I'm very confused about the attached questions on geometric buckling of a cylinder and a sphere. For this question I'm not given a value for the extrapolated distances (for R' and H') so I simply put the...

Homework Statement From Misner, Thorne and Wheeler's text Gravitation (MTW), exercise 3.15: Show that, if F is the EM field tensor, then ##\nabla \cdot *F## is a geometric, frame-independent version of the Maxwell equation...

Jack draws rings of rhombuses about a common centre point. All rhombuses have the same side length. Rhombuses in the first, or inner, ring are all identical. Each rhombus has a vertex at the centre and each of its sides that meet at the centre is shared with another rhombus. They all have the...

Reading Chandrasekhar's The mathematical theory of black holes, I reached the point in which the Newman-Penrose GR formalism is explained. Actually I'm able to grasp and understand the usage of null tetrads in GR, but The null tetrads used in this formalism, are very special, since are made by...

Given that the sum of the first n terms of series, s, is 9-32-n Find the sum of infinity of s. Do I use the formula S\infty = \frac{a}{1-r}?

Given that the sum of the first n terms of series, s, is 9-32-n show that the s is a geometric progression. Do I use the formula an = ar n-1? And if so, how do I apply it?

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...

Dear Everybody, I need some help with find M in the definition of the convergence for infinite series. The question ask, Prove that for $-1<r<1$, we have $\sum_{n=0}^{\infty} r^n=\frac{1}{1-r}$. Work: Let $\sum_{n=0}^{k} r^n=S_k$. Let $\varepsilon>0$, we must an $M\in\Bbb{N}$ such that $k\ge...

A geometric sequence has an initial value of 25 and a common ratio of 1.8. Write a function to represent this sequence . Find the 23rd term. My Effort: The needed function is a_n = a_1•r^(n-1), n is the 23rd term, r is the common ratio and a_1 is the initial value. a_23 = 25•(1.8)^(23 - 1)...

Solve for $\theta$ in the diagram below.

I used to think it was called Zeno's tower, but then realized I probably called it that because it reminded me of his paradox. I have been unable to find this shape on the internet, although I saw a small steel tower outside Stonybrook using this geometry. I have attached an image of the basic...

I am trying to understand the geometric intuition of the above equation. ##\rho(\tau)## represents the rank of the linear transformation ##\tau## and likewise for ##\rho(\tau\sigma)##. ##Im(\sigma)## means the image of the linear transformation ##\sigma## and lastly, ##K(\tau)## is the kernel of...

Hi all. So to start I'll say I'm just dealing with functions of a real variable. In my linear algebra courses one thing was drilled into my head: "Algebraic invariants are geometric objects" So with that in mind, is there any geometric connection between two orthoganal functions on some...

Hello all, I have a question related to geometric probability. I think I solved it, but not sure, would appreciate your opinion. We are given a round table with a radius of 50cm. At the center of this table there is another circle, with a radius of 10cm. A coin with a radius of 1cm is thrown...

β is vehicle sideslip (angle between velocity and vehicle forward vector) ψ the angle between the trajectory tangent and vehicle forward vector. I have this equation that says Vx * k * (cos ψ - tan β * sin ψ) where k is trajectory...

Homework Statement Calculate the geometric phase change when the infinite square well expands adiabatically from width w1 to w2.Homework Equations Geometric phase: \gamma_n(t) = i \int_{R_i}^{R_f} \Bigg< \psi_n \Bigg | \frac{\partial \psi_n}{\partial R} \Bigg > dR Infinite square well wave...

Homework Statement A person with a near point of 100 cm , but excellent distant vision, normally wears corrective glasses. But he loses them while traveling. Fortunately, he has his old pair as a spare. If the lenses of the old pair have a power of +2.55 diopters , what is his near point...