Step 1: Understanding the Concept:
This problem focuses on power-based relationships. In competitive exams, recognizing perfect squares and cubes is a critical skill for speed. When a number in the ratio is significantly larger than its counterpart, you should immediately check if it represents a square (\(n^2\)), a cube (\(n^3\)), or a value very close to them.
Step 2: Key Formula or Approach:
The number 1331 is a famous mathematical constant—it is the cube of 11. We need to find how the base (11) relates to the original number (13).
The pattern is: \[ \text{Number}_2 = (\text{Number}_1 - 2)^3 \]
Step 3: Detailed Explanation:
1. Decoding the First Pair (13 : 1331):
We identify that \( 1331 = 11 \times 11 \times 11 \).
The relationship between the base 11 and the first term 13 is:
\[ 13 - 2 = 11 \]
Therefore, the logic applied is to subtract 2 from the first number and then find the cube of the result.
2. Applying the Logic to the Second Pair (17 : ?):
The third number is 17.
Step 1: Subtract 2 from 17:
\[ 17 - 2 = 15 \]
Step 2: Calculate the cube of 15:
\[ 15^3 = 15 \times 15 \times 15 \]
First, find the square of 15: \( 15 \times 15 = 225 \).
Then, multiply the square by 15: \( 225 \times 15 = 3375 \).
3. Evaluating the Options:
Comparing our result (3375) with the given choices:
(A) 2642 (Incorrect)
(B) 1453 (Incorrect)
(C) 3375 (Matches our calculation)
(D) 1829 (Incorrect)
Step 4: Final Answer:
The logic follows the rule \( (n-2)^3 \). Applying this to 17 gives 3375.