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: Understanding the Concept:
A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio, denoted by \(r\).
Step 2: Key Formula or Approach:
If the terms are \(a_1, a_2, a_3, ...\), then the common ratio \(r = \frac{a_{n+1}}{a_n}\). Also, any term can be expressed as \(a_n = a_1 \cdot r^{n-1}\). We can use these relationships to solve for the unknowns.
Step 3: Detailed Explanation:
The given geometric progression is \(x, 6, y, 54, 162\).
Let the terms be \(a_1=x, a_2=6, a_3=y, a_4=54, a_5=162\).
Method 1: Find r, then x and y
We can find the common ratio \(r\) using two consecutive known terms, \(a_4\) and \(a_5\).
\[ r = \frac{a_5}{a_4} = \frac{162}{54} = 3 \] Now that we know \(r=3\), we can find \(x\) and \(y\).
For \(x\): \(a_2 = a_1 \cdot r \implies 6 = x \cdot 3 \implies x = \frac{6}{3} = 2\).
For \(y\): \(a_3 = a_2 \cdot r \implies y = 6 \cdot 3 \implies y = 18\).
Now, calculate the required ratio \(\frac{y}{x}\):
\[ \frac{y}{x} = \frac{18}{2} = 9 \] Method 2: Using GP term relationships
We want to find \(\frac{y}{x}\). In our sequence, \(y = a_3\) and \(x = a_1\).
Using the formula \(a_n = a_1 \cdot r^{n-1}\), we can write:
\[ a_3 = a_1 \cdot r^{3-1} = a_1 \cdot r^2 \] Substituting \(y\) and \(x\):
\[ y = x \cdot r^2 \] Therefore, \(\frac{y}{x} = r^2\).
We still need to find \(r\). As in Method 1, \(r = \frac{162}{54} = 3\).
So, \(\frac{y}{x} = (3)^2 = 9\).
Step 4: Final Answer:
The value of \(\frac{y}{x}\) is 9. Therefore, option (C) is the correct answer.