How many three-digit numbers ABC in which A, B, and C are each digits, satisfy the equation
2B = A + C
2B = A + C
This implies that the digits A, B, and C form an A. P.
For common difference (d) = 0,
ABC can be
111, 222, 333, 444, 555, 666, 777, 888, and 999
Total 9 digits with d = 0
For common difference (d) = 1,
ABC can be
123, 234, 345, 456, 567, 678, 789, 987, 876, 765, 654, 543, 432, 321, and 210
Total 15 digits with d = 1
Now, for d = 2
ABC can be
135, 246, 357, 468, 579, 975, 864, 753, 642, 531, and 420
Total 11 digits with d = 2
For d = 3,
ABC can be
147, 258, 369, 963, 852, 741, and 630
Total 7 digits with d = 3
For d = 4,
ABC can be
159, 951, 840
Total 3 digits with d = 4
Therefore, total 45 such numbers are there which satisfies 2B = A + C
Option D is the answer.
