<s>[INST] Why does this pattern occur: $123456789 \times 8 + 9 = 987654321$ I came across the following:
$\begin{align} 1 \times 8 + 1 &= 9 \\ 12 \times 8 + 2 & = 98 \\ 123 \times 8 + 3 & = 987 \\ 1234 \times 8 + 4 & = 9876 \\ 12345 \times 8 + 5 & = 98765 \\ 123456 \times 8 + 6 & = 987654 \\ 1234567 \times 8 + 7 & = 9876543 \\ 12345678 \times 8 + 8 & = 98765432 \\ 123456789 \times 8 + 9 & = 987654321. \\ \end{align}$
I'm looking for an explanation for this pattern.  I suspect that there is some connection to the series $\frac{1}{(1 - x)^2} = 1 + 2x + 3x^2 + \cdots$.  
This post asks the same question but has no answers posted.
 [/INST] If I consider the equations you provide with your "ideas so far":
\begin{align}
1 \times 9 + 1 &= 10 \\
12 \times 9 + 2 & = 110 \\
123 \times 9 + 3 & = 1110 \\
 \vdots\\
123456789 \times 9 + 9 & = 1111111110, \\
\end{align}
The first equation being true, this system is equivalent to the system composed of  their successive differences all of them having the common pattern :
$$\underbrace{11...1}_{k \ \text{digits}} \times 9 + 1 = 10^k$$
which is an (almost) evident fact.
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