Sequence and Series

Concept:The product of $n$ geometric means inserted between two numbers equals the $n$th power of the single geometric mean between them.Explanation:Let the two numbers be $a$ and $b$ with $b > a > 0$.Insert $n$ geometric means $G_1, G_2, \dots, G_n$ between $a$ and $b$.These form a geometric progression: $a, G_1, G_2, \dots, G_n, b$.Let the common ratio be $r$. Then $G_k = a r^k$ for $k = 1, 2, \dots, n$ and $b = a r^{n+1}$.Product of the $n$ means: $\displaystyle \prod_{k=1}^{n} G_k = a^n r^{1+2+\cdots+n} = a^n r^{\frac{n(n+1)}{2}}$.The single geometric mean between $a$ and $b$ is $\sqrt{ab} = \sqrt{a \cdot a r^{n+1}} = a r^{\frac{n+1}{2}}$.Its $n$th power: $( \sqrt{ab} )^n = a^n r^{\frac{n(n+1)}{2}}$.Both expressions are equal, so the statement is true.Answer:True