Why is $9 \times 11{\dots}12 = 100{\dots}08$?

Question

While I was working on Luhn algorithm implementation, I discovered something unusual.

$$ 9 \times 2 = 18 $$ $$ 9 \times 12 = 108 $$ $$ 9 \times 112 = 1008 $$ $$ 9 \times 1112 = 10008 $$

Hope you can observe the pattern here.

What to prove this? What is it's significance?

Answer 1

The repunit, $R_k = \overbrace{111\ldots 111}^{k \text{ ones}}$ , can be written as $R_k = \dfrac{10^k-1}{9}$

Your nice pattern corresponds to $9\times (R_k+1) = (10^k-1)+9 = 10^k+8$

Answer 2

Here's an idea.

This pattern is easier to understand: $$ 9 \times 11 = 99$$

$$9 \times 111 = 999$$

$$9 \times 1111 = 9999$$

Then see what happens when you add 9 to both sides.

Answer 3

$$9 \times 11\cdots12 = 9 \times (11\cdots 11+1) = 99\cdots99 + 9.$$

Answer 4

Write $1...12$ as $$\overline{1...(\times\ n)...1} +1=\frac{10^n-1}{9}+1$$

Hence $$9\times (\frac{10^n-1}{9}+1)=10^n-1+9=10^n+8$$

Answer 5

One logic behind.

$9 \times (0 + 2) = 9 \times 0 + 9 \times 2 = 18$

$9 \times (10 + 2) = 9 \times 10 + 9 \times 2 = 108$

$9 \times (110 + 2) = 9 \times 110 + 9 \times 2 = 1008$

So on.