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# It looks simple but can you explain it?

How would you describe the sequence of numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, ...?

Nota Bene: You're not allowed use a computer program or code to describe it.

Note by Hobart Pao
2 years, 5 months ago

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Hint: Apply the general formula for $$C_{10}$$ for Champernowne constant. Then 10 raise to the power of respective digits, then mod 10 answer. The full working is very long and ugly.

- 2 years, 5 months ago

So,

$T_n = \lfloor 10^{n} \times C_{10} \rfloor \pmod{10}$

- 2 years, 5 months ago

AKA

$\large T_k = \left \lfloor 10^k \times \displaystyle \sum_{m=0}^{\infty} \sum_{n=10^{m-1}}^{10^m - 1} \dfrac {n}{10^{m(n - 10^{m - 1} + 1) + 9 \sum_{l=1}^{m-1} 10^{l-1} l}} \right \rfloor \pmod{10}$

- 2 years, 5 months ago

Comment deleted Oct 25, 2015

Isn't $$C_{10} = 0.123456789101112131415 \ldots$$? I don't see why there'd be a problem with the relation.

- 2 years, 5 months ago

Oh wait. You're right. I was thinking of something completely different. Thanks! =D

But since OP wanted the full formula, you might want to type out that nasty double summation in full. HAHA!

- 2 years, 5 months ago

Written out the LaTeX hell. Had to put it in large so it could be read.

- 2 years, 5 months ago

My eyes is in pain just by staring at it!

- 2 years, 5 months ago

Thanks guys! I love what you guys did! I thought it'd be simpler because this was on an entrance exam for a specialized middle school.

- 2 years, 4 months ago