Find two geometric means between $\frac { 1 } { 32 }$ and $\frac { 1 } { 4 }$ .
A) 2 and 16
B) $\frac { 1 } { 2 }$ and $\frac { 1 } { 16 }$
C) 8 and 16
D) $\frac { 1 } { 8 }$ and $\frac { 1 } { 16 }$

Question

Quizplus · Accepted Answer

To find the geometric means between $\frac{1}{32}$ and $\frac{1}{4}$, we use the formula for the nth root of a product, where the number of terms (including the two given terms) is 4. The common ratio (r) can be found by $\sqrt[n-1]{\frac{a_n}{a_1}}$, where $a_n = \frac{1}{4}$ and $a_1 = \frac{1}{32}$. Here, $n-1 = 3$, so $r = \sqrt[3]{\frac{1/4}{1/32}} = \sqrt[3]{8} = 2$. Therefore, the two geometric means are $\frac{1}{32} \times 2 = \frac{1}{16}$ and $\frac{1}{16} \times 2 = \frac{1}{8}$, making the correct answer $\frac{1}{8}$ and $\frac{1}{16}$.

Find Two Geometric Means Between 132\frac { 1 } { 32 }321​

Find Two Geometric Means Between $\frac { 1 } { 32 }$