Proving a Simple Formula for n = 2,4,6,8 Using Mathematical Induction

duki
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Homework Statement



Conjecture a simple formula for n = 2,4,6,8

Homework Equations



Kind of...
Tn = (1-\frac{1}{n^2})(1-\frac{1}{(n-1)^2})... to n=1

The Attempt at a Solution



The pattern is 3/4, 5/8, 7/12, 9/16
I know that the top is increasing by 2 and the bottom is increasing by 4... but I don't know how to put that into a formula to which I can prove.

Thanks for any help!
 
Last edited:
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oooooh snap, how about this:

Tn = \frac{n+1}{n*2} for all even n?
 
Why not?
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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