Let a1, a2, a3, ... be a non-constant arithmetic sequence. Suppose that a4, a7, a12 form a geometric sequence. If a7 = 30, what is $a_1$?

Answer 1

a₄· a₁₂ ·a₄ = a₇² = 900; (a₁ + 3d)(a₁ + 11d) = (a₁ + 6d)² = 900; a₁ + 6d = 30; a₁ = 30 - 6d;

(30 - 6d + 3d)(30 - 6d + 11d) = 900; 3(10 - d)·5(6 + d) = 900; - d² + 4d + 60 = 60; d(d - 4) = 0; d = 0 (we reject.

because prgressioon not constant) or d = 4. Then a₁ = 30 - 24 = 6

Answer 2