Geometric Definition and 813 Threads

I would like to find a nice formula for \sum_{k=0}^{n - 1}ar^{4k}. I know that \sum_{k=0}^{n - 1}ar^{k} = a\frac{1 - r^n}{1 - r} and was wondering if there was some sort of analogue.

Hi guys, It's been a while since high school, and now I'm faced with a problem I need to solve in a few days (attached). Would someone please help me through that? I would really appreciate support.

Problem 1 A flat screen TV is place on a wall in a room. A lens of focal length 50cm is placed between the television and the opposite wall so that a sharp image with one quarter of that of the area of the television is produced on the opposite wall. Answer: Magnification =...

Homework Statement Well, the problem is to find the area of: {(x,y), -(pi/2) <= x <= Pi/2, 1/2< y <=cosx} The Attempt at a Solution Well, I know that for an angle of 60 degrees, the cosine is 1/2. So I guess that the limits of my integral will be from 0 to Pi/3. But I'm getting confused...

Homework Statement Homework Equations a(1-r^[n+1])/(1-r) The Attempt at a Solution So I wrote it as e^(-iNz) [1 + e^(iz) + e^(2iz) + ... + e^(2iNz)] Let r = e^(iz), a=e^(-iNz) a [1 + r + r^2 + ... + r^(2N)] From here I'm not sure what to do. I tried letting n=2N, and...

Just a little help understanding results obtained. I have found the closed form of a sequence, but am a little unsure if there is a right way or can select either way of using the terms to create the explicit formula. I have found the common difference from the terms, which is 1.2, in my...

Homework Statement a) P(X=x)=pq^x,\,x\geq 0 Find the PGF. b) P(X=x)=pq^{|x|},\,x\,\epsilon\,\text{Z} Find the PGF. 2. The attempt at a solution a) G_X(s)=E(s^X)=\displaystyle\sum_{x\geq 0}pq^x s^x=p\displaystyle\sum_{x\geq 0}(qs)^x=\frac{p}{1-qs} b) Not sure about this one... Is it: as...

Let r = √(x^2+y^2+z^2) One can easily show that \nablar= \vec{r}/r. But I'm having a hard time understanding what this means geometrically - who can help? :)

I'm interested in the proper way to give a mathematical definition of a certain geometric property exhibited by certain maps from points to sets. Consider mappings from a n-dimensional space of real numbers P into subsets of an m-dimensional space S of real numbers. For a practical...

I have performed numerous calculations of dot products throughout my math courses, but I think I lack a fundamental understanding of what it actually means, beyond the abstract way I have been taught to deal with them. I know the definitions (it's the inner product, or the projection of A on to...

I am a beginner in theory of GR and am trying to understand it better. I have a problem with understanding tensors. I got the algebriac idea, incliding covariance, contravariance and transformations etc of tensors. But not the geometric. Tensors are abstract but can I not have geometric...

Homework Statement Find y component of vector C from its length and the angle it makes with the x axis, that is, from geometry. Express the y component of vector C in terms of C and \phi. Homework Equations Vector addition using geometry: 1) C = \sqrt{A^{2}+B^{2}-2ABcos(c)}...

Hello everyone on these forums. :) If you would, please consider the 3-vector function r(t) = <f(t),g(t),h(t)>. What sort of geometric meaning can be assigned to the following integral? \int_a^b \vec{r}(t) dt = \left\langle \int_a^b f(t) dt, \int_a^b g(t) dt, \int_a^b h(t) dt\right\rangle...

Until today I learned in geometric optics that Object distance +ve for real object else -ve Image distance +ve for real image else -ve Radius of curvature +ve for if light comes to the surcace from the side lying center of curvature else -ve On the basis of this the lens formula...

Homework Statement we had a a function on a graph of f(x)=1/x and then we are suposed to find the area of a triangle where the tangent line is the hypontenuse, and the x and y-axis are the base and hight...i found f'(x)= -1/x^2 from here i used the formula y-yo=x0(x-x0) and got that the x...

Homework Statement I already counted V_{0}=-1 and q=\frac{1}{3} given: V_{n}=1-\frac{2}{U_{n}} Homework Equations count: \sum_{k=0}^{n}V_{k} The Attempt at a Solution i counted the sum and i got : ((\frac{1}{3})^{n+1}-1)(\frac{2}{3}) is that correct?

i guys, I'm stuck on wording of a homework assignment and thought you might be able to help me. There are several questions... Consider the geometric series: (Sum from k=0 to infinity) of ar^k and consider the repeating decimal .717171717171 for these problems: Question 1: Find a formula...

1. Homework Statement you are geometrically diluting/mixing 0.1 g of powder A with 100g of powder B, how many times do you have to mix the 2 together to finish the process? *each time you can only mix an equal portion of powder B to what you currently have mixed. Eg. 1:1 2:2 4:4 3...

Homework Statement you are geometrically diluting/mixing 0.1 g of powder A with 100g of powder B, how many times do you have to mix the 2 together to finish the process? *each time you can only mix an equal portion of powder B to what you currently have mixed. Eg. 1:1 2:2 4:4 The...

Hi, I'm trying to come up with a probability for a game I play with a friend of mine. In the game, units "attack" by rolling six-sided dice; either 2 or 4 sides of the die count as a "hit" when rolled, depending on certain circumstances. The specific situation I am trying to figure out the...

Homework Statement I have a question with asks to solve a differential equation via power series and I've done everything up to finding the recurrence relation which is a_{n+2} = -\frac{a_{n}}{n+2} Given the initial conditions a_{o} = 1 and a_{1} = 0 I'm trying to simplify the series into a...

Hello everyone! My question is twofold. Firstly, how do I solve for term numbers in a geometric sequence and secondly, how do I algebraically solve for variables that are exponents? Homework Statement Given the following geometric sequences, determine the number of terms, n. t1=5 r...

Homework Statement x,y, z are vectors in R^n. We have the equation: ax +by +cz, where a,b,c are constants such that a+b+c=1, and a,b,c>=0 What is the geometric interpretation of the equation? Homework Equations sv + tu, where u,v are vectors in R^n and s,t are constants such that...

Homework Statement I am trying to prove the sum of a geometric series, but one of the steps involves deriving this result: \lim_{n\to\infty}r^{n}=0 so that you can simplify the sum of a geometric series, where I have got to this stage: S_{\infty} = \frac{a(1-r^{\infty})}{1-r}...

Homework Statement The sum of an infinite geometric sequence is 131/2, and the sum of the first three terms is 13. Find the first term. Homework Equations S∞ = a/(1-r) Sn = a-arn/(1-r) The Attempt at a Solution a/(1-r) = 131/2 a-ar3/(1-r) = 13 2a = 27-27r ...... 1 a-ar3 =...

I've seen a number of books and articles touting Geometric Algebra as an important new area of math that will have large application to physics. Is there anything to these claims? Is it worth studying for a physics student?

Homework Statement Find the sum of Ʃ(3→∞) 3(.4)^(n+2) Homework Equations Sum of Geometric Series = ao/(1 - r), ao=3, r = .4 = 2/5 The Attempt at a Solution I thought that I could use the definition of the sum of a geometric series (above) to determine the sum of this equation...

Say we have an eigenvalue \lambda and corresponding eigenvectors of the form (x,x,2x)^T. What is the geometric multiplicity?

Dot product proof question? Hi, I'm having trouble understanding the proof of the dot product in three dimensions (not using the cosine rule approach). Here's what I have for the 2D proof: u = u1 i + u2 j v = v1 i + v2 j u.v = u1v1 + u2v2 u.v = |u| |v| cos(θ) => u1v1 + u2v2 = |u| |v|...

Homework Statement A 20 × 20 matrix C has characteristic polynomial (λ^2 − 4)^10. It is given that ker(C−2I), ker (C − 2I)^2, ker (C −2I)^3 and ker (C −2I)^4 have dimensions 3,6,8,10 respectively. It is given that ker (C + 2I), ker (C +2I)^2, ker (C +2I)^3 and ker (C +2I)^4 have di- mensions...

I've been browsing through MTW recently and I found something that puzzles me: They claim that if you have two form, call it \mathbf{T}, it's value, say \mathbf{T}(\mathbf{u} , \mathbf{v} ) can be represented geometrically as follows: take two vectors \mathbf{u} and \mathbf{v}; the surface...

Bill Alsept started a thread raising the general question---do cosmic models with regularly repeating big bangs conflict with thermodynamics' 2nd Law? (The law to the effect that, where it can be defined, entropy does not decrease, or does so only by rare accident, at irregular intervals if at...

Homework Statement I feel bad asking another question after I just asked one yesterday, but I'm really close this time, I think! I have: \sum_{n=2}^{\infty}\frac{n^2-n}{2^n} And need to find the sum. Homework Equations \sum_{n=1}^{\infty}nx^{n-1}=\frac{1}{(1-x)^2} The Attempt at a...

Consider the following experiment: a coin that lands heads with probability p is flipped once; if on this first flip it came up H, it is then repeatedly flipped until a T occurs; else, if on the first...

This has been a curiosity of mine lately. I am wondering about what makes an algebra person an algebra person. I know geometers(at least it seems like it) seem to have a keen ability of spatial visualization. What characterizes the abilities of an algebra person? To clarify, I'm not just talking...

Can anyone help me answer this question? " Every exponential function is a geometric progression but not every geometric progression is an exponential function. Explain."

Geometric series problem urgent Homework Statement Calculate the geometric series of Ʃfrom n=1 to infinity of 1/n Homework Equations The Attempt at a Solution I don't know how to start solving, how can I solve this? I have test about this tomorrow I really need some help please.

Statistics: geometric distribution "proof" problem Homework Statement If Y has a geometric distribution with success probability p, show that: P(Y = an odd integer) = \frac{p}{1-q^{2}} Homework Equations p(y)=p(q)^{2} The Attempt at a Solution p(1)=pq^0 p(3)=pq^2 p(5)=pq^4...

Hi, All: Just curious to know if there is an interpretation for lower cohomology that is as "nice", as that of the lower fundamental groups, i.e., Pi_0(X) =0 if X is path-connected (continuous maps from S^0:={-1,1} into a space X are constant), and Pi_1(X)=0 if X is...

Hello, I have the following equation in x and y: xy - \sqrt{(x^2+a^2)(y^2+c^2)} = -\frac{1}{a^2}-\frac{1}{c^2} where the quantities a2 and c2 are given real constants, and I have to find real values for x, and y such that the equation above is always satisfied. Actually, I know that the...

For a homogeneous system of 3 equations in 3 unknowns (geometrically this is 3 planes in space all containing the origin) describe the relationship between the (three) geometric possibilities for the solution set and the number of free variables (non pivots) in RREF(A) where A is the...

Homework Statement If O is any point inside a triangle ABC, prove that BA + AC > BO + OC. Homework Equations The Attempt at a Solution Any hints? Thanks...

I need to find the solution to the geometric series expansion of the form... \sumn^2*x^n , for n=0,1,2,... most resources I've found only have answers for n*x^n or n*x^(n-1). I have no idea how to calculate this, so I was wondering if there's a book out there that has massive lists of...

I am currently doing a course on Computer Graphics Algorithms. This involves lot of matrix transformations i.e. for eg - rotating co-ordinates, translating, reflecting etc. I am solving the problems on paper using a calculator, but I need some software which will help me verify the solution...

We're covering this in my History of Mathematics class. I'm not entirely sure what they're asking.

Homework Statement Q.: Show that if log a, log b and log c are three consecutive terms of an arithmetic sequence, then a, b and c are in geomtric sequence. Homework Equations Un = a + (n - 1)d and Sn = \frac{a(r^n - 1)}{r - 1} The Attempt at a Solution Attempt: Consider...

Homework Statement Q.: The sum of the first five terms of a geometric series is 5 and the sum of the next five terms is 1215. Find the common ratio of this series. Homework Equations Sn = \frac{a(r^n - 1)}{r - 1} The Attempt at a Solution a + ar + ar^2 + ar^3 + ar^4 = 5 ar^5 +...

Homework Statement Q.: The numbers \frac{1}{t}, \frac{1}{t - 1}, \frac{1}{t + 2} are the first, second and third terms of a geometric sequence. Find (i) the value of t, (ii) the sum to infinity of the series. Homework Equations S\infty = \frac{a}{1 - r} The Attempt at a...

Homework Statement Q.: A geometric series has first term 1 and common ratio \frac{1}{2}sin2\theta. Find the sum of the first 10 terms when \theta = \frac{\pi}{4}, giving your answer in the form h - \frac{1}{2^k}, where h, k \in N. Homework Equations Sn = \frac{a(1 - r^n)}{1 - r}, when...

Homework Statement (1) \frac{1}{(1+x^{2})}+\frac{1}{(1+x^{2})^{2}}+...+\frac{1}{(1+x^{2})^{n}} The Attempt at a Solution (2)...