If 6 @ 4 @ 7 = 101 and 2 @ 5 @ 11 = 150, then following the same pattern, what is the value of A in the expression A @ 8 @ 9 = 289?

Introduction / Context:
This is a pattern recognition problem using a custom three argument operator denoted by the symbol @. The operator combines three numbers to produce a result. We are given two examples that show how the operator behaves and must infer the rule in order to find the missing value A in the third expression. Such questions test the ability to see numerical patterns and generalize them correctly.

Given Data / Assumptions:

• 6 @ 4 @ 7 = 101.
• 2 @ 5 @ 11 = 150.
• A @ 8 @ 9 = 289 and we need to find the value of A.
• The same operator rule is used in all three expressions.
• The numbers appear in the same left to right order inside each expression.

Concept / Approach:
The natural idea is to test simple patterns involving the three numbers. One strong candidate is the sum of squares because the results 101 and 150 are close to typical sums of squared numbers. By checking whether each result equals the sum of the squares of the three arguments, we can verify the pattern quickly. If it fits both given examples, we can safely apply it to the third expression to solve for A.

Step-by-Step Solution:
Step 1: Test the expression 6 @ 4 @ 7. Compute 6², 4² and 7². Step 2: 6² = 36, 4² = 16 and 7² = 49. Step 3: Sum these squares: 36 + 16 + 49 = 101, which matches the given result. So 6 @ 4 @ 7 equals the sum of the squares of 6, 4 and 7. Step 4: Test the second example 2 @ 5 @ 11. Compute 2² = 4, 5² = 25 and 11² = 121. Step 5: Add them: 4 + 25 + 121 = 150. This matches the second given result as well. Step 6: From both examples we infer that a @ b @ c = a² + b² + c² for any three numbers a, b and c in this context. Step 7: Apply the rule to A @ 8 @ 9. The expression equals A² + 8² + 9². Step 8: We are told A @ 8 @ 9 = 289. So A² + 8² + 9² = 289. Step 9: Compute 8² = 64 and 9² = 81, so 64 + 81 = 145. Step 10: Substitute into the equation: A² + 145 = 289, hence A² = 289 − 145 = 144. Step 11: The square root of 144 is 12, and A is a positive integer in the options, so A = 12.

Verification / Alternative check:
We can confirm the logic by substituting A = 12 back into the pattern. Compute 12² + 8² + 9² = 144 + 64 + 81 = 289, which matches exactly with the given result. No other option gives a square whose sum with 145 is 289, so the solution is unique among the answer choices.

Why Other Options Are Wrong:
If A were 5, 8 or 17, the values of A² would be 25, 64 and 289 respectively. Adding 145 to any of these does not give 289. For example, 25 + 145 = 170 and 64 + 145 = 209, both far from 289. Only 144 + 145 equals 289. Thus options 5, 8 and 17 do not satisfy the fundamental relation determined by the pattern.

Common Pitfalls:
A frequent mistake is to try random combinations like products plus sums or differences without first testing a simple and natural idea such as the sum of squares. Another common error is to forget that the same rule must work for both given examples, not just one of them. Always confirm that a suspected pattern fits every provided case before using it to compute a missing value.

Final Answer:
Using the rule that a @ b @ c equals the sum of the squares of the three numbers, the missing value is 12.