Repunit Expressed using Power of 10

Theorem

The repunit number $R_n$ can be expressed as:

$R_n = \dfrac {10^n - 1} 9$

$\ds \dfrac {10^n - 1} 9$

$=$

$\ds \dfrac {10^n - 1} {10 - 1}$

$\ds $

$=$

$\ds \sum_{k \mathop = 0}^{n - 1} 10^k$

$\ds $

$=$

$\ds 1 + 10 + 100 + \ldots + 10^{n - 2} + 10^{n - 1}$

The result follows from the Basis Representation Theorem.

$\blacksquare$

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): $1,111,111,111,111,111,111$