Question

A three digit number with distinct digits has its digits in increasing geometric progression. if the middle digit is doubled and the hundredth digit is tripled, then resulting digits would be in arithmetic progression. find its unit digit.

Answer 1

Suppose, the digits are a/r, a, ar because the digits are in increasing geometric progression.

Now, according to the question:

3a/r, 2a, ar will be in A.P.

Hence, 2a-3a/r=ar-2a

r^2-4r+3=0

(r-1)(r-3)=0 we get,

r=1,3

r=1 is not possible because, it will make all the digits equal.

So, we take r=3 and we get:

a/3, a, 3a

Here, we can take a=3 randomly and we will get our G.P. as 1, 3, 9.

Therefore, the unit digit will be 9 considering G.P. is increasing from hundredth place to unit place.