Question

The three geometric means between 4 and 324 are

• A. 12, 46, 108
• B. 12, 36, 128
• C. 14, 36, 108
• D. 12, 36, 98
• E. 12, 36, 108

Hint:

Logic Tip: The middle term of three geometric means ($g_2$) is exactly the single geometric mean between the boundary numbers. $\sqrt{4 \cdot 324} = \sqrt{1296} = 36$. Knowing this instantly narrows down the correct answer.

Answer 1

Answer: (E)

Concept:
Inserting $n$ geometric means between two numbers $a$ and $b$ creates a Geometric Progression consisting of $n + 2$ total terms, where $a$ is the first term and $b$ is the last term.
Step 1: Define the sequence terms.
[cite_start]We need to insert 3 geometric means ($g_1, g_2, g_3$) between $4$ and $324$[cite: 120]. The sequence is: $4, g_1, g_2, g_3, 324$. This is a G.P. with a total of $5$ terms. First term ($a$) = 4 Fifth term ($a_5$) = 324
Step 2: Determine the common ratio (r).
Using the formula for the $n$-th term $a_n = ar^{n-1}$: $$a_5 = a \cdot r^4$$ $$324 = 4 \cdot r^4$$ Divide by 4: $$r^4 = 81$$ Taking the fourth root gives: $$r = 3 \text{ or } r = -3$$ Looking at the multiple-choice options, all listed sequences are positive, so we proceed with $r = 3$.
Step 3: Calculate the geometric means.
Multiply successively by the common ratio $r = 3$: $$g_1 = a \cdot r = 4 \cdot 3 = 12$$ $$g_2 = g_1 \cdot r = 12 \cdot 3 = 36$$ $$g_3 = g_2 \cdot r = 36 \cdot 3 = 108$$ [cite_start]The three geometric means are $12, 36, \text{ and } 108$, which matches Option E[cite: 129, 130].