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Prove that the sequence $\ {1, 11, 111, 1111,.\ldots\}$ will contain two numbers whose difference is a multiple of $2017$ I have been computing some of the immediate multiples of $2017$ to see how their congruence classes look like, but i am not really sure where that is taking me. How do i calculate this sum in terms of 'n' I know this is a harmonic progression, but i can't find how to calculate the summation of it

Also, is it an expansion of any mathematical function The factor 1/3 attached to the $n^3$ term is also obvious from this observation. The formal moral of that example is that the value of 1i 1 i depends on the branch of the complex logarithm that you use to compute the power You may already know that 1= e0+2kiπ 1 = e 0 + 2 k i π for every integer k k, so there are many possible choices for log(1) log (1).

How do i convince someone that $1+1=2$ may not necessarily be true

I once read that some mathematicians provided a very length proof of $1+1=2$