Repunit Expressed using Power of 10
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Theorem
The repunit number $R_n$ can be expressed as:
- $R_n = \dfrac {10^n - 1} 9$
Proof
| \(\ds \dfrac {10^n - 1} 9\) | \(=\) | \(\ds \dfrac {10^n - 1} {10 - 1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^{n - 1} 10^k\) | Sum of Geometric Sequence | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1 + 10 + 100 + \ldots + 10^{n - 2} + 10^{n - 1}\) |
The result follows from the Basis Representation Theorem.
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1,111,111,111,111,111,111$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1,111,111,111,111,111,111$