Derivation of Product of First n Terms of Geometric Progression
The product of the first $n$ terms of a Geometric Progression is given by the following:
Given the first term $a_1$ and last term $a_n$:
Given the first term $a_1$ and the common ratio $r$
Derivation of Formulas
$P_n = a_1 \times a_2 \times a_3 \times \ldots \times a_{n - 1} \times a_n$
$P_n = a_n \times a_{n - 1} \times a_{n - 2} \times \ldots \times a_2 \times a_1$
$P_n \times P_n = (a_1 \times a_2 \times a_3 \times \ldots \times a_{n - 1} \times a_n)(a_n \times a_{n - 1} \times a_{n - 2} \times \ldots \times a_2 \times a_1)$
${P_n}^2 = (a_1 \times a_n)(a_2 \times a_{n - 1})(a_3 \times a_{n - 2}) \, \cdots \, (a_{n - 1} \times a_2)(a_n \times a_1)$
$a_2 \times a_{n - 1} = a_1 \times a_n$
$a_3 \times a_{n - 2} = a_1 \times a_n$
$\cdots \,$ and so on
Hence,
${P_n}^2 = (a_1 \times a_n)(a_1 \times a_n)(a_1 \times a_n) \, \cdots \, (a_1 \times a_n)(a_1 \times a_n)$
${P_n}^2 = (a_1 \times a_n)^n$
$P_n = (a_1 \times a_n)^{n/2}$ ← Formula
Replace $a_n$ by $a_1 r^{n - 1}$
$P_n = (a_1 \times a_1 r^{n - 1})^{n/2}$
$P_n = ({a_1}^2 \, r^{n - 1})^{n/2}$
$P_n = {a_1}^n \, r^{n(n - 1)/2}$ ← Formula
Another Way to Derive $P_n = {a_1}^n \, r^{n(n - 1)/2}$
$a_{n - 1} = a_1 \, r^{n - 2}$
$a_{n - 2} = a_1 \, r^{n - 3}$
$\cdots$
$a_1 = a_1 \, r^{n - n} = a_1 \, r^0$
$P_n = a_n \times a_{n - 1} \times a_{n - 2} \times \dots \times a_1$
$P_n = a_1 \, r^{n - 1} \times a_1 \, r^{n - 2} \times a_1 \, r^{n - 3} \times \ldots \times a_1 \, r^0$
There are $n$ factors of $a_1$ and the powers of $r$ is in the form of arithmetic progression
$P_n = {a_1}^n \, r^{(n/2) \, [ (n - 1) + 0 ]}$
$P_n = {a_1}^n \, r^{n(n - 1)/2}$ ← Formula
See also the derivation for the sum of GP.