Arithmetic progression (AP) constraints: The 4th term of an AP is 37, and the 6th term is 12 more than the 4th term. What is the sum of the 2nd and 8th terms?
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A80
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B86
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C74
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D92
Answer
Correct Answer: 86
Explanation
Introduction / Context:This problem tests basic AP formulas: the k-th term t_k = a + (k − 1)d. Knowing two terms lets us find the common difference and first term, then any other term or combination such as t2 + t8.Given Data / Assumptions:
- t4 = a + 3d = 37.
- t6 = a + 5d = t4 + 12 = 49.
- Find t2 + t8.
Concept / Approach:Solve for d from the difference between t6 and t4, then back-solve a. Compute t2 and t8 and add. Alternatively, use an identity to avoid computing a explicitly.Step-by-Step Solution:
From t6 − t4 = (a + 5d) − (a + 3d) = 2d = 12 ⇒ d = 6.From t4: a + 3d = 37 ⇒ a + 18 = 37 ⇒ a = 19.t2 = a + d = 19 + 6 = 25.t8 = a + 7d = 19 + 42 = 61.Sum t2 + t8 = 25 + 61 = 86.Verification / Alternative check:Identity: t2 + t8 = (a + d) + (a + 7d) = 2a + 8d = 2(a + 3d) + 2d = 2*37 + 12 = 86. Confirms without computing a separately.Why Other Options Are Wrong:
- 80, 74, 92: Do not satisfy the AP relationships given by t4 and t6.
Common Pitfalls:Misreading “6th is 12 more than 4th” or mixing term indices. Always convert to algebraic equalities and proceed systematically.
Final Answer:
86