<s>[INST] What is the greatest value... The sum of the first three terms of a geometric progression is 21, and the sum of their squares is 1281. What is the greatest value can be the sum of the cubes?
 [/INST] Call the terms $\dfrac{b}{r}$, $b$, and $br$.  Square the sum. We get 
$$\frac{b^2}{r^2}+b^2+b^2r^2 +2b\left(\frac{b}{r}+b+br\right).$$
Thus $441= 1281+2b(21)$. It follows that $b=-20$. The rest is routine.
If we want to do the rest prettily, use
$$\frac{1}{r^3}+r^3=\left(\frac{1}{r}+r\right)^3-3\left(\frac{1}{r^2}+r^2\right).$$
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