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I came across this sequence in the paper:

109080, 12123, 1350, 153, ?

What could be the next number? And what is the pattern that connects these numbers?

Dividing a term by the next term leads to a number which consecutively decreases from 8.99 for the first term to 8.8 for the third term Is that of any use?

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The answer is

20

Because

If you compute one term minus nine times the next one, you always get $-27$. In other words: $$x_{i+1}=\frac{x_i+27}9 \,\, \left(\text{or $\frac{x_i}9+3$}\right)$$ Note that $20$ is the last integer in this sequence, because it falls under $9$ at the next step.

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    $\begingroup$ Thanks for the answer! $\endgroup$ – Vin Aug 11 at 14:03
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    $\begingroup$ @Vin You're welcome. Actually, the last observation you made in the puzzle is the reason why I tried that. Think about what is natural and close to 8.99... $\endgroup$ – Arnaud Mortier Aug 11 at 14:09
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    $\begingroup$ Or $x_n = (133*9^n+27)/8$ for $n=4,3,2,1,0$ $\endgroup$ – Florian F Aug 13 at 20:36

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