The sum of 225 consecutive positive integers only has digits 1 or 0. What is the smallest possible value of the sum?
The sum of the sequence must be divisible by 225, or rather multiples of 225 that end in at least 2 zeros. So, will list multiples of 225: 225, 450, 675, 900. Since ALL the digits of the sum of this sequence are only 1's and zeros, then the sum must have a minimum of 9 ones in order to be divisible by 9.Or: 111111111. But we want multiple of 900, so we just add 2 zeros at the end of 9 ones: 11,111, 111,100. This is the sum of your arithmetic sequence.
Now, we can use the formula for the sum of an AP to find out the FIRST term:
11,111,111,100 = 225/ 2 * [2*F + 1*(225 - 1)]
= 112.5 * [2*F + 224]
= 225F + 25,200
225F =11,111,085,900
F = 11,111,085,900 / 225
F =49,382,604 and this is your FIRST term of the sequence. If you start with this term and add 1 to each subsequent term for 225 terms until you get the LAST term, or 49,382, 828, and sum them all up, you should get:11,111,111,100. And that is the END of this question! I wonder who thought of this?.
