Find a three digit number such that its digits are in geometric progression and the digits of number obtained from it by subtracting 400 form an arithmetic progression.

Hint: let the three digit number be abc. and let r be the common ratio. Then b=ra, c=r2a.
Now for A.P. the number after subtracting 400 will become (a-4)bc. Let h be the common difference then, b = a-4+h and c = a-4 +2h. Now equate the b and c and get the relation between r and h. Also note that the possible values of r = 0,1,2 or 3. and for h = 0,1 ,2,3 or 4. Now take the different cases for r and h. You will see that only for r=0 and h=0 will be possible. Now you can easily use these values to find the three digit number. Thanks.