Identify the odd triple: Three options form an arithmetic progression with common difference 2 (even-step AP or near-AP). Choose the triple that does NOT form an exact AP.
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A20, 16, 18
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B14, 11, 13
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C18, 14, 16
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D16, 12, 14
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ENone of these
Answer
Correct Answer: 14, 11, 13
Explanation
Introduction / Context:Arithmetic progressions (AP) are sequences where consecutive terms differ by a constant. Here, the intended AP has common difference 2 after arranging terms in ascending order. One triple fails to be an exact AP.
Given Data / Assumptions:
- Triples: (20, 16, 18), (14, 11, 13), (18, 14, 16), (16, 12, 14).
- We may reorder triples to test for AP.
Concept / Approach:Sort each triple and check whether middle − smallest = largest − middle (both should equal 2). Any deviation marks the odd triple.
Step-by-Step Solution:
(20, 16, 18) → sorted: 16, 18, 20 → differences 2 and 2 → AP.(14, 11, 13) → sorted: 11, 13, 14 → differences 2 and 1 → not an AP.(18, 14, 16) → sorted: 14, 16, 18 → differences 2 and 2 → AP.(16, 12, 14) → sorted: 12, 14, 16 → differences 2 and 2 → AP.Verification / Alternative check:An AP with common difference 2 must satisfy smallest + largest = 2 * middle. For 11, 13, 14: 11 + 14 = 25 ≠ 26, confirming the mismatch.
Why Other Options Are Wrong:They all form neat APs with common difference 2 upon sorting.
Common Pitfalls:Forgetting to sort before checking differences, or miscomputing one of the differences and rejecting a valid AP.
Final Answer:14, 11, 13 is the odd triple.