The sum of three numbers in AP is 27. If 1 is added to the first number, 2 to the second, and 3 to the third, the new numbers are in GP. Find the numbers.

The sum of three numbers in AP is 27.

Let us assume the numbers are a-d, a , a+d

Where d is the common difference.

Sum a-d+a+a+d = 27

3a = 27

a = 9

If 1 is added to the first number, 2 to the second, and 3 to the third, the new numbers are in GP.

So, the new numbers are 

9-d+1, 9+2, 9+d+2

10-d , 11, 11+d

Checking options:

A) 6, 9, 12 when we add numbers it becomes 7, 11, 15 (Not in GP)

B) 5, 9, 13 when we add numbers it becomes 6, 11, 16 (Not in GP)

C) 10, 9, 8 when we add numbers it becomes 11, 11, 11 (In GP)

D) 4, 9, 14 when we add numbers it becomes 5, 11, 17 (Not in GP)

Hence, answer is option C.