USSR OLYMPIAD PROBLEM

[MATHPRITOM]

Find the last 100 digits of the number N=$1+50+50^2+50^3+...+50^{999}$.

[Moon]

You get a recurring string of $41$ digit number, which is precisely \[020408163265306122448979591836734693877551\]
I think wolfram alpha can sometimes do better math than we can.
http://www.wolframalpha.com/input/?i={5 ... 1}%2F{50-1}

[Masum]

$\varphi(49)=42$, so in the decimal expansion of $\frac1{49}$ has period $42$, not $41$ and for this we should not use Wolfram. Now using this as a hint, try to show that these digits should be $0$.

[Moon]

Oh, I miscalculated the period. BTW how can all these digits be zero, when it is relatively prime to 10. Did I miss something?

[Masum]

Think think!!