Happy New Year!
Algebra
Level
3
2016,1,2016,1,1,2016,1,1,1,2016,1,1,1,1,2016,...
Find the sum of the first 2016 digits
The answer is 128961.
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1 solution
Let´s consider every 2016 the first column of a triangles line:
2016, 1
2016, 1, 1
2016, 1, 1, 1
2016, 1, 1, 1, 1
2016, ....
We will have n+1 numbers in each line, where n is the number of the line, but, since the problem limits the number of terms on the sequence, we can have a unfinished line, let´s call it line number N. To find N we will create a number p, that is equal to the number of terms "missing" to complete line number N, and put all this and a equation:
2+3+4+...+N+(N+1)-p = 2016 1+2+3+4+...+N = 2016+(N-p)
Knowing that 2016+(N-p) is a sum of an arithmetic progression, with N terms that add up to a number smaller than 2016, we can guess number like N=8 and find out n1 and n2, such as: n1=<2016=<n2.
To N=8: s = 1+2+3+4+5+6+7+8 = (1+8)x(8/2) = 36
To N=16: s = 1+2+3+4+5+6+7+8+...+16 = (1+16)x(16/2) = 136
To N=32: s = 1+2+3+4+5+6+7+8+...+32 = (1+32)x(32/2) = 528
To N=64: s = 1+2+3+4+5+6+7+8+...+64 = (1+64)x(64/2) = 2080
To N=63: s = 1+2+3+4+5+6+7+8+...+63 = (1+63)x(63/2) = 2016
Wait, if N=63 gives us s = 2016, we have exactly 63 lines on our triangle, and the sum of its terms, is the sum of its lines terms, that is another arithmetic progression, 2017+...+2016+63 = 2017+...+2079 = (2017+2079)x(63/2) = 129024.
That means that either my first answer is wrong, or this one is wrong, or both are wrong, hahaha, but I would bet on 129024, because I got this answer doing things more calmly. ;)