Is the following number (with the added conditions taken into account) always irrational? And how would one go about proving this? Heres the number and the conditions:

Is the number

always irrational? Given these conditions:

for some numbers

and

for some numbers

and

There, I think I've covered all the proper conditions. But if there is one I missed, I'm sure you catch the drift of the exact question I am asking, and can probably guess whatever the missing condition is, not that there is a missing condition, just considering the possibility. Anyway, theres the question. Also, I understand the possibility that there is a simple example that shows the number

to be rational in some obvious case, and if there is such a case, then this question is solved simply. And if your answer is yes, I"d be interested in seeing a proof, or atleast the "gist" of the proof. Thanks in advance