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.

#Algebra

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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.

Pi Han Goh - 5 years, 7 months ago

So,

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

Sharky Kesa - 5 years, 7 months ago

AKA

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

Sharky Kesa - 5 years, 7 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.

Hobart Pao - 5 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$