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April 12th, 2012, 06:35 PM  #1 
Senior Member Joined: Apr 2012 Posts: 112 Thanks: 0  Number theory and finding the domain and range!
Hi I have three questions, two on number theory and the other on finding range and domain. 1.) Find the largest domain possible for the function f(x)> (2)/(2x^2 6x+5) and find the range corresponding to this domain. I would like to know, is there any easy, shortcut method to do this? I have tried, it took a long time and my answer came out incorrect. What I did was to manually sub in numbers for x and then derive a conclusion from there. It is tedious and I am aware that I cant do this in an exam. How would you solve it? 2.) a,b and c are consecutive terms of both an AP and GP. Therefore a=b=c I don't understand why a=b=c. Surely the value must increase or decrease from a to c? Unless they're talking about the difference, but I don't see that in the question. 3.) 1+((3/2)x)+((3/2)x)^2+((3/2)x)^3+... I know that the common ratio is r=(3/2)x The question asks, write down the range of values for x for the series to converge. The answer was 1<r<1 The rest of the steps I know how to continue, but I do not understand why for the series to converge, r has to be 1<r<1? I had expected 1<x<1 Also, what is the value of r happen if the series is to diverge? Someone told me r>0, is that right? Thank you so much! 
April 12th, 2012, 07:08 PM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  Re: Number theory and finding the domain and range!
1.) Equate the denominator to zero to find what values (if any) to exclude from the set of reals for the domain: Thus, we find no real numbers to exclude. Thus the domain is: Now to find the range, the function will have its maximal value when the denominator is at its minimum, which will occur at the axis of symmetry: At this value of x the denominator is and thus Since the function has a horizontal asymptote of , the range is then: [attachment=0:44r3687v]range.jpg[/attachment:44r3687v] 2.) For a, b and c to be consecutive terms in an arithmetic progression, we require: For a, b and c to be consecutive terms in a geometric progression, we require: Thus, we have: Hence: 3.) Check out this article on the condition for convergence: http://en.wikipedia.org/wiki/Geometric_series 
April 12th, 2012, 08:41 PM  #3 
Senior Member Joined: Apr 2012 Posts: 112 Thanks: 0  Re: Number theory and finding the domain and range!
Thank you so much!! 

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