AP constraints: The 9th term exceeds the 5th term by 32, and their sum is 114. Find the 8th term of the AP.

Introduction / Context:
Two linear constraints on terms of an AP allow solving for the first term a and common difference d. Once a and d are known, any term is immediate.

Given Data / Assumptions:

• t_n = a + (n−1)d.
• t_9 − t_5 = 32; t_9 + t_5 = 114.

Concept / Approach:
Write both constraints in a and d, solve the linear system, then compute t_8.

Step-by-Step Solution:
t_9 − t_5 = (a+8d) − (a+4d) = 4d = 32 ⇒ d = 8.t_9 + t_5 = (a+8d) + (a+4d) = 2a + 12d = 114 ⇒ 2a + 96 = 114 ⇒ a = 9.t_8 = a + 7d = 9 + 56 = 65.

Verification / Alternative check:
Compute t_5 = 9 + 4*8 = 41 and t_9 = 9 + 8*8 = 73; difference 32 and sum 114 confirm.

Why Other Options Are Wrong:
60, 63, 68 are not equal to a + 7d with a = 9, d = 8.

Common Pitfalls:
Using n instead of (n−1) in t_n; arithmetic slips when adding the equations.

Final Answer:
65