Calculating The Nth Rational Number

[moyo]

Hallo

If we specify a particular method for mapping the natural numbers to the rationals, could we also specify a "distance" between two consecutive terms in some general way. Also are we able to calculate the nth term in such a progression perhaps incorporating this distance function somehow within its expression.

 

[MountEvariste]

Your question is not very clear. What are you referring to when you say "two consecutive terms", "nth term" and "progression"?

 

[HOI]

Hallo

If we specify a particular method for mapping the natural numbers to the rationals, could we also specify a "distance" between two consecutive terms in some general way. Also are we able to calculate the nth term in such a progression perhaps incorporating this distance function somehow within its expression.

If "we specify a particular method for mapping the natural numbers to the rationals" is the key. If we have some function $$\{a_n\}$$ such that to every natural number n, we have a rational number $$a_n$$ and every rational number is on that list, then, for any n we could determine $$a_{n+1}- a_n$$. Whether there would be any reasonable formula for that function of n depends on the mapping. And asking whether "we able to calculate the nth term in such a progression" is asking whether there exist a reasonable function describing that progression.

Since the rational numbers are countable, such progressions exist but whether or not there exist reasonable formulas for calculating them depends on the progression.

 

[moyo]

Thanks for the response :)

https://www.geeksforgeeks.org/nth-rational-number-in-calkin-wilf-sequence/

I found that, its a whole algorithm for calculating the nth term.