It isn't clear to me whether you mean a binary number with a pattern that is repeated some finite number of times, or one that is infinitely repeated. I will assume you mean infinitely repeated, as in, say,
.11010110101101011010.....
which repeats infinitely every five binary digits. Let x be the value of the first pattern (.11010 in the example) and n be the number of binary digits in it. Then the value of the infinitely repeating pattern would be:
y = x/(1-2^(-n))
In the above example it would be y = (26/32)/(1-2^(-5)) = 26/31 = .8387096774
There is a similar formula for one repeated a finite number of times.
