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?

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:

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: