Geometric progression — determine the first term: In a finite GP, the sum of the first and the last terms is 66, and the product of the second and the second-last terms is 128. Find the first term.

Introduction / Context:
Let the first term be a, common ratio r, and number of terms n. Denote t = r^(n−1). The given conditions relate a and t without needing n explicitly.

Given Data / Assumptions:

• a + a t = 66 ⇒ a(1 + t) = 66.
• (a r) * (a r^(n−2)) = a^2 r^(n−1) = a^2 t = 128.

Concept / Approach:
Eliminate r and n using t. Solve the system a(1+t) = 66 and a^2 t = 128.

Step-by-Step Solution:
a = 66 / (1 + t)a^2 t = 128 ⇒ (66^2 t) / (1 + t)^2 = 128Solve: 4356 t = 128(1 + 2t + t^2) ⇒ 32 t^2 − 1025 t + 32 = 0t = 32 or t = 1/32 ⇒ a = 66/(1+32) = 2, or a = 66/(1 + 1/32) = 64

Verification / Alternative check:
Both a = 2 and a = 64 satisfy the relations for appropriate r and n symmetric about the sequence ends.

Why Other Options Are Wrong:
Singletons 64 or 2 exclude the other valid branch; 32 is not a valid first term from the derived quadratic.

Common Pitfalls:
Mixing “sum of first and last” with total GP sum; attempting to solve for r and n unnecessarily.

Final Answer:
64 or 2