Question

If the numbers \( x, 6, y, 54, 162 \) are in geometric progression, then \( \dfrac{y}{x} \) is equal to

• A. \( 3 \)
• B. \( 6 \)
• C. \( 9 \)
• D. \( 12 \)
• E. \( 18 \)

Hint:

In GP problems, always use the ratio between known consecutive terms first. It simplifies the entire calculation quickly.

Answer 1

The Correct Option is C

Step 1: Understand the concept of Geometric Progression (GP).
In a geometric progression, each term is obtained by multiplying the previous term by a constant ratio \( r \).
So the sequence can be written as: \[ x, xr, xr^2, xr^3, xr^4 \]

Step 2: Match the given terms with the GP form.
Given sequence: \[ x, 6, y, 54, 162 \] Comparing with GP: \[ x = x,\quad 6 = xr,\quad y = xr^2,\quad 54 = xr^3,\quad 162 = xr^4 \]

Step 3: Find the common ratio \( r \).
Using the last two terms: \[ r = \frac{162}{54} = 3 \]

Step 4: Find the value of \( x \).
From: \[ 6 = xr \] Substitute \( r = 3 \): \[ 6 = 3x \Rightarrow x = 2 \]

Step 5: Find the value of \( y \).
We know: \[ y = xr^2 \] Substitute values: \[ y = 2 \cdot 3^2 = 2 \cdot 9 = 18 \]

Step 6: Compute \( \dfrac{y}{x} \).
\[ \frac{y}{x} = \frac{18}{2} = 9 \]

Step 7: Final conclusion.
Thus, the required value is: \[ \boxed{9} \] Correct option is: \[ \boxed{(3)\ 9} \]