(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let [tex]A = (a_1, . . . , a_{2008}) \in \Re^{2008}[/tex], where [tex]a_i = \frac{1}{2^{i}}[/tex] for each [tex]i = 1, . . . , 2008[/tex]. Find the distance from the point [tex]A[/tex] to the origin. Please express your answer in the form [tex]\sqrt{\frac{a}{b}}[/tex] where [tex]a, b[/tex] are integers.

2. Relevant equations

Extended distance formula in [tex]\Re^{n}[/tex]

3. The attempt at a solution

Using what I knew about the distance from a point to the origin in two and three dimensions, I deduced the distance is

[tex]\sqrt{\sum_{i=1}^{2008} a_i^2} = \sqrt{\sum_{j=1}^{2008} \frac{1}{2^{2j}}} = \sqrt{\sum_{k=0}^{2007} \frac{1}{4}\frac{1}{4^k}} = \left( \frac{1}{4} \right) \left( \frac{1 - \left( \frac{1}{4} \right)^{2008}}{1 - \frac{1}{4}} \right)[/tex]

where the last equality is achieved through the finite geometric series formula. Now I did simplify that expression so that the numerator and denominator contained products of integers but when a problem normally asks for integers a and b, don't they usually want them in decimal notation? I mean I know that the numerator and denominator will be integers but the expression is too big to multiply out. Thus, I'm not quite sure of my solution to this problem. Thanks for your help.

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# Homework Help: Distance formula in higher dimensions.

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