Question 388635
hi,
when you look at positive three-digit whole numbers of which the middle digit is the average of the other two digits. one example is 741. how many of these numbers exist?


average = 1 --> sum 1st,3rd add to 2 --> 1 + 1, 2 + 0 --> 2 numbers
ave = 2 --> sum add to 4 --> 1 + 3, 2 + 2, 3 + 1, 4 + 0 --> 4 numbers
ave = 3 --> sum add to 6 --> 1 + 5, 2 + 4, 3 + 3, 4 + 2, 5 + 1, 6 + 0 --> 6
ave = 4 --> sum add to 8 --> 1 + 7, 2 + 6, 3 + 5, 4 + 4, 5 + 3, 6 + 2,
............................   7 + 1, 8 + 0 --> 8 numbers
ave = 5 --> sum add to 10 --> 1 + 9, 2 + 8, 3 + 7, 4 + 6, 5 + 5, 6 + 4,
............................   7 + 3, 8 + 2, 9 + 1 --> 9
ave = 6 --> sum add to 12 --> 3 + 9, 4 + 8, 5 + 7, 6 + 6, 7 + 5, 8 + 4,
............................   9 + 3 --> 7
ave = 7 --> sum add to 14 --> 5 + 9, 6 + 8, 7 + 7, 8 + 6, 9 + 5 --> 5
ave = 8 --> sum add to 16 --> 7 + 9, 8 + 8, 9 + 7 --> 3 numbers
ave = 9 --> sum add to 18 --> 9 + 9 --> 1 number
2 + 4 + 6 + 8 + 9 + 7 + 5 + 3 + 1
appears to be the sum of the digits from 1 to 9
sum digits = n(n+1)/2 = 9(10)/2 = 90/2 = 45 numbers
111, 210
123, 222, 321, 420
135, 234, 333, 432, 531, 630
147, 246, 345, 444, 543, 642, 741, 840
159, 258, 357, 456, 555, 654, 753, 852, 951
369, 468, 567, 666, 765, 864, 963
579, 678, 777, 876, 975
789, 888, 987
999