What is the largest three-digit integer whose digits are distinct and form a geometric sequence?
I want to find a three-digit number of the form abc, where a, b, and c are distinct digits and b/a = c/b, which is the definition of a geometric sequence. I also want to maximize this number, so I want to make a as large as possible, which means a = 9. Now, since b/a = c/b, I can cross-multiply and get b^2 = ac. I need to find two digits b and c such that b^2 = 9c and b and c are distinct. I can try different values of b and see if they work. If b = 9, then c = 9, which is not distinct from a, so that doesn't work.
[*, If b = 8, then c = 64/9, which is not a digit, so that doesn't work either. If b = 7, then c = 49/9, which is not a digit, so that doesn't work. If b = 6, then c = 36/9 = 4, which is a digit and distinct from a and b, so that works. Therefore, the largest three-digit number whose digits are distinct and form a geometric sequence is 964. 