Mathematics

Use Bertrand's Postulate to show that every positive integer n with n >= 7 is the sum of distinct primes. I know that Bertrand's Postulate states that for every positive integer n with n > 1, there is a prime p such that n < p < 2n. So, in our case since n >= 7 > 1 we can deduce that n < p < 2n That's pretty much all I can say looking at the postulate itself. What else?

Hi all! I'm supposed to prove the following: Suppose that m is a positive real number. Show that Sigma(j^m), j runs from 1 to n is O(n^(m+1)). So we have: Sigma(j^m), j runs from 1 to n = 1^m + 2^m + 3^m + . . . + n^m I think we should use induction on m here to prove this. Basis Step: m = 1 This is definitely true for m = 1 because we have Sigma(j^1), j runs from 1 to n = n(n+1)/2 which is O(n^2) Now, O(n^2) = O(n^(1+1) = O(n^(m+1) Inductive Step: Assume Sigma(j^m), j runs from 1 to n is O(n^(m+1)) to be true. That is, it is true for m = n. (Inductive Hypothesis) Then must show it is also true for m = n + 1: …

Hi all! I'm supposed to use Euclid's proof that there are infinitely many primes to show that the nth prime p_n does not exceed 2^(2^(n-1)) whenever n is a positive integer. Conclude that when n is a positive integer, there are at least n+1 primes less than 2^(2^n). Any help will be greatly appreciated. Thanks!

I have a vague recolection from Pre-Calculus of learning about unit vectors... Vector "i" and "j"... one of which goes from (0,0) to (0,1), the other (0,0) to (1,0). Does any this ring a bell to anyone? and if so, can you tell me which one is which? I can't seem to find it. I think "i" is (0,0) -> (0,1), and "j" is (0,0) -> (1,0). And, is there one of these for the thrid dimention? (eg, from (0,0,0) -> (0,0,1))

I'm just wondering if anyone knows any decent books (not magazines) for puzzles. I am not interested in sudoku, word searches, crosswords etc... I prefer maths, geometry, logic and science puzzles. Especially ones that are challenging (but not impossible i.e. PhD standard!!!!) Websites are good, but I'd prefer books... Thanks

This is my first assignment of my first math class or my first year of univercity and they said i should have learned this in grade 11, (too long ago to recall really) so i want to make sure i got them right (which i doubt!) here's how it is, i will give the question and my answer and if i am wrong please tell me. 1) [math]\frac{5x^4}{\sqrt{x^3-6}}+x\sqrt{x^3-6}[/math] solution: [math]\frac{6x^4-6x}{\sqrt{x^3-6}}[/math] 2) [math]\frac{\frac{x^2-x-6}{4x-12}}{\frac{2x+6}{x^2+5x+6}}[/math] solution: [math]\frac{(x+2)^2}{8}[/math]

Today my maths teacher showed us how we can expand [math](x+1)^n[/math] using pascals triangle. For instance, to find [math](x+1)^4[/math], we know that the fourth layer of pascals triangle is 1,4,6,4,1 and these numbers come into the awnser:[math]x^{4}+4x^{3}+6x^{2}+4x+1[/math], the other bits to be put in are fairly simple. This was pretty impressive but if the exponent were 15 I wouldn't really want to draw up a 15 layered pascal triangle, I'm to lazy. So my question is: is there a way I can work out the [math]n^{th}[/math] layer of Pascals triangle without working down the whole thing?

Finding limits of infinite series is possible but I don't know how to calculate the limit of a multiplicative series. I put the problem I have on a website including the background to the problem, would be interested in any insights or alternative ways of doing the sum.... http://www.geocities.com/mobiusmaths/infpoly.htm

This is a question from my school text book:- The line y = 3x - 5 meets the x-axis at the point M. The line y = -2/3x + 2/3 meets the y axis at the point N. Find the equation of the line joining the points M and N. Write your answer in the form ax + by + c = 0. For point M i got the coordinate (5/3, 0) and for point N (0, 2/3) I next worked out the gradient of the line connecting M and N to be -2/5 and tried using y - y1 = m(x - x1) to get the equation for the line eventually getting to 6x - 15y - 3 = 0, however the book lists the answer as 6x + 15y - 10 = 0. Is my answer the correct answer and the book's answer wrong, or did i mess up in my working out …

a vector is given by [math]\vec{v}=\sum_{j=1}^{n}a_j{\vec{i_j}}[/math] where i is the unit vector for a given dimension and n is the number of dimensions of the space in which the vector exists. the magnitude ([math]|\vec{v}|[/math]) is given by [math]|\vec{v}|^2=\sum_{j=1}^{n}a_j^2[/math]. the unit vector in the direction of the vector v is given by [math]\vec{u}=\frac{\vec{v}}{|\vec{v}|}[/math]. if theta is the angle between vectors A and B, then [math]{\vec{A}}*{\vec{B}}={|\vec{A}|}{|\vec{B}|}{\cos{{\theta}}}=\sum_{j=1}^{n}a_jb_j[/math]. [math]\cos{\theta}=\sum_{j=1}^{n}\frac{a_jb_j}{|\vec{A}||\vec{B}|}[/math]. the projection of A on B is [math]Proj_{\vec{B}}{\vec{A}}…

This is from a 6 part physics homework problem, I got part a-e, but part f (likely to be my grade ) is what I'm having some trouble with: A mass m slides down a frictionless incline and sticks to a spring (spring constant k ), initiating simple harmonic motion. f) What is the angular frequency of the SHM? I got [math](k/m)^{1/2}[/math] as the solution (someone please tell me how to get the Latex radical )

The arithmetic sequence a, a+d, a+2d, a+3d, ...,a+(n-1)d has the following properties. When the first, third, fifth, and so on are added to and including the last term, the sum is 320 When the first, fourth, seventh and so on, terms are added, up to and including the last term, the sum is 224 What is the sum of the whole sequence?

Can someone explain to me how to solve for x in this example? ln(x-2)=3 Also, is there a difference between ln, lg, and log or do they all mean the same thing? Thanks

can someone please introduce me to Hyperplanes. thank you

I need to calculate the distance between postcodes to find out how far people travel.I have obtained details such as latitude and longtitude for each postcode and know it is possible to do this but how? example of the information held postcode x (in metres)y latitude longitude AB10 392900 804900 57.135 -2.117 AB14 383600 801100 57.101 -2.27 EX35 272400 148800 51.224 -3.828 EX36 274300 125700 51.016 -3.792 Anyone have any ideas? Thanks dave

Example: x = √3x+7-1 x+1 = √3x+7 (x+1)² = (√3x+7)² x²+2x+1 = 3x+7 Question: Why +2x? x²-6-x = 0 Question: Why -6-x? x = -2 OR x = 3 Question: How can you see the answer will be -2 or 3?

Hello! What I'd like to ask is, whether there is a mathematical theory of consciousness in development or any promising serious research projects in describing conscious process mathematically or physically with sufficient mathematical background

The famous Last Theorem of Fermat (proven by Andrew Wiles in the early 1990's) http://en.wikipedia.org/wiki/Andrew_Wiles says There are no positive integer solutions for [MATH]n>2[/MATH] to the equation [MATH]z^n=x^n+y^n[/math] We will not prove this theorem here (Wiles method is very advanced). But we will play around with the implications a bit. Something fun for us mortals to do si assume the negative of FLT and then start messing around a bit and see what happens. It remains entertaining only because some people believe Fermat actually solved it with only his knowledge of number theory and that someday the "real" proof will re-emerge. It see…

Ok this is my problem. I dont even know how to use this program but i need to plot graphs with it for a project. Where are you supposed to input all the data and stuff. eg. i have this y=x2 <--squared i dont know any button for it. Ok so y=x2 and then x has diffrernt values like -5 then the next bit -4 so on and so on..so can anyone help me or send me any links i really need help!

hey , i'm having a bit of trouble with this linear transformation problem. i am not really sure what or how they want me to show? so yeah, does anyone have some suggestions or advice? Cheers Sarah

Howdy. I've been working through this site called Calculus On the Web or COW. It's been a great site so far, but I have a question on the section entitled 'Trigonometric Identities' At the top of the page it shows the situation where x = cos a y = sin a Then at the bottom of the same page, in an example, it says that side B = tan a I don't understand this. In the example at the top, isn't side B = y coordinate = sin a how can side B = tan a? Unfortunately there is not a direct link to the page I'm talking about. You must go here : Calculus On the Web and then click on : Precalculus Book Functions Trigonometry Trig …

If I have x number of balls in a lottery, how many times will I have to draw a ball at random and replace it before I draw a ball that has already been drawn once before? I developed an algorithm to solve this, I think it works, not entirely sure. But I was wondering how to solve it algebraicly, here is my algorithm: t = number of balls in lottery x = 1 n = 0 while (x>0.5) { n = n + 1 x = x * (t - n + 1) / t } Answer = n

i was trying to make a trig chart for the complex plane and accindently found 0>1. at pi/4, the hypotenuse is 0, but the real leg is 1. how is that? imaginary trig must be fun. edit: at [math]\theta=\frac{\pi}{4}[/math], i got: sin=undefined cos=undefined tan=i csc=0 sec=0 cot=-i it is in the first quadrant and cot is negative...something is wonky

Well I know you guys closed out one of these but the concept is interesting. Ordinal numbers and cardinal numbers I know a bit but not everything and cardinal numbers are part of one of the most important math discoveries of the 20th century Paul Cohen's result on the continuum hypothesis. Any takers?

Hey guys! (I posted the same thread in physicsforums. To grasp a wider audience I decided to post it here too.) They say the best way to develop and train problem solving skills is by creating one's own problems. I'm still a "passive" problem solver but I'd like to learn solving problems in an "active" way. Could you give me some advice, name some guidelines on how to create new, own problems? Thanks in advance.