Step 1: Understanding the Concept:
This is a numerical analogy problem. We need to identify the relationship between the first pair of numbers (50 and 65) and then apply the same relationship to the second pair to find the missing number.
Step 2: Key Formula or Approach:
The relationship might be arithmetic (addition, subtraction) or based on a more complex pattern, often involving squares or cubes of numbers.
Let's analyze the first pair, 50 and 65.
- The difference is \(65 - 50 = 15\). Applying this to the second pair, \(290 + 15 = 305\), which is not an option.
- Let's look for a pattern involving squares. The numbers are close to perfect squares.
- \(50 = 7^2 + 1\)
- \(65 = 8^2 + 1\)
This reveals a pattern: the relationship is from \(n^2+1\) to \((n+1)^2+1\).
Step 3: Detailed Explanation:
Analyze the first pair:
- The first number is \(50 = 49 + 1 = 7^2 + 1\).
- The second number is \(65 = 64 + 1 = 8^2 + 1\).
The pattern is that if the first number is \(n^2+1\), the second number is \((n+1)^2+1\). For the first pair, \(n=7\).
Apply the pattern to the second pair:
- The given number is 290. We need to see if it fits the form \(k^2+1\).
- \(290 = 289 + 1 = 17^2 + 1\).
This matches the pattern, with \(k=17\).
- The missing number should be \((k+1)^2+1\).
- \((17+1)^2 + 1 = 18^2 + 1\).
- Calculate \(18^2\): \(18 \times 18 = 324\).
- The missing number is \(324 + 1 = 325\).
Step 4: Final Answer:
The missing number is 325, which corresponds to option (D).