1430.11 – The Mean 9's

Consider these numbers: 9, 99, 999, 9999, 99999, 999999, 9999999, 99999999, 999999999

If you write down the arithmetic mean (the average) of these numbers you will write every digit except which one?

  1. 0

  2. 2

  3. 4

  4. 6

  5. 8

Solution

The arithmetic mean is:

9+99+⋯+9999999999=99+999+⋯+9999999999=1+11+⋯+111111111=123456789\begin{align*}\dfrac{9 + 99 + \cdots + 999999999}{9} &= \dfrac{9}{9} + \dfrac{99}{9} + \cdots + \dfrac{999999999}{9} \\[1em]&= 1 + 11 + \cdots + 111111111 \\[0.5em]&= 123456789\end{align*}

So the answer is (a). Cute, no?