If 1 6 , , are in G.P., then the general solution of is

Step-by-step solution

Given the progression a = 1 6 , b = , c = are in geometric progression, the condition holds: b^2 = ac . Substitute the expressions: ( )^2 = ( 1 6 ) ( ) . Since = , ( ^2 = 1 6 . ) Multiplying both sides by ( 0 ): ( ^3 = 1 6 ^2 . ) Using ^2 = 1 - ^2 : ( ^3 = 1 6 (1 - ^2 ). ) Multiply by 6 and rearrange: (6 ^3 + ^2 - 1 = 0. ) Let x = : (6x^3 + x^2 - 1 = 0. ) Check rational roots: x = 1 2 satisfies the equation, so it is a root. Factor the cubic: (2x - 1)(3x^2 + 2x + 1) = 0 . The quadratic factor 3x^2 + 2x + 1 = 0 has