<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] This is something I noticed but I'm still thinking about if it means anything:
$$\boxed{1\cdot8+1=9}\\\downarrow$$
$$10\cdot8+10=90$$
$$10\cdot8+18=98$$
$$(10+2)\cdot8+2=98$$
$$\boxed{12\cdot8+2=98}\\\downarrow$$
$$120\cdot8+20=980$$
$$120\cdot8+27=987$$
$$(120+3)\cdot8+3=987$$
$$\boxed{123\cdot8+3=987}\\\downarrow$$
$$1230\cdot8+30=9870$$
$$1230\cdot8+36=9876$$
$$(1230+4)\cdot8+4=9876$$
$$\boxed{1234\cdot8+4=9876}\\\downarrow\\\cdot\\\cdot\\\cdot$$
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