Mathematics

In a geometric progression, the terms are symmetrically arranged around the middle—the first term pairs with the last, the second with the second-to-last, and so on. The key insight is that each such pair multiplies to the same value: $a \cdot b$. Since there are $n$ terms total, the product of all $n$ terms can be regrouped as $n$ copies of this pairing, giving you $P = (ab)^{n/2}$, which immediately yields $P^2 = (ab)^n$.

In a geometric progression, the terms are symmetrically arranged around the middle—the first term pairs with the last, the second with the second-to-last, and so on. The key insight is that each such pair multiplies to the same value: $a \cdot b$. Since there are $n$ terms total, the product of all $n$ terms can be regrouped as $n$ copies of this pairing, giving you $P = (ab)^{n/2}$, which immediately yields $P^2 = (ab)^n$.