Distance formula in higher dimensions.

[snipez90]

Homework Statement

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

Homework Equations

Extended distance formula in $$\Re^{n}$$

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

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

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.

 

[tiny-tim]

say it in binary!

… 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.

Hi snipez90!

Normally, yes …

but I think 4²⁰⁰⁸ - 1 is an exception!

hmm …

if you really want to impess your professor

(thought not necessarily favourably …)

you could write the answer out in full …

in binary! ​

​

 

[snipez90]

Haha well to be honest this isn't my pset. I'm itching for college to begin and thought I would prepare by trying the problem sets of friends who have already begun.

I passed along the hint though, thanks for the confirmation :-D.