Selection of Terms of a G.P.

When we work with a geometric progression, we often need to pick a specific set of terms that satisfy a given condition — like a fixed sum or product. The trick is to choose the terms symmetrically around the middle term so that the common ratio cancels out neatly in calculations. For an odd number of terms, say three, we pick them as $\frac{a}{r}$, $a$, $ar$; for five, we use $\frac{a}{r^2}$, $\frac{a}{r}$, $a$, $ar$, $ar^2$, and so on. This symmetrical selection keeps t …