Since the numbers are in the ratio 1:3:8, then k:3k:8k where k is
not zero, represents all triples of numbers in the ration 1:3:8

So let the three numbers be k,3k, and 8k

Then the terms k, 3k, and 8k+7 form a geometric sequence.

The common ratio =  = 

Therefore   = 
            = 
Multiply through by 3k  
            
           9k = 8k+7
            k = 7

Then the numbers are  k,3k, and 8k or 7,21,56

Checking: adding 7 to the 3rd term gives the sequence 7,21,63

That checks, since 7*3=21 and 21*3=63

Edwin