If the coefficient of x 15 in the expansion of a x 3 + 1 b x 1 3 15 is equal to the coefficient of x - 15 in the expansion of a x 1 3 - 1 b x 3 15 , where a and b are positive real numbers, then for each such ordered pair a , b :

Step-by-step solution

( r + 1 ) t h term in the expansion of a x 3 + 1 b x 1 3 15 is T r + 1 = C r 15 a x 3 15 - r 1 b x 1 / 3 r ⇒ T r + 1 = C r 15 a 15 - r x 45 - 10 r 3 1 b r For the coefficient of x 15 in a x 3 + 1 b x 1 / 3 15 : 45 - 10 r 3 = 15 ⇒ 30 = 10 r 3 ⇒ r = 9   Coefficient of x 15 = C 9 15 a 6 b - 9 ( r + 1 ) t h term in the expansion of a x 1 / 3 - 1 b x 3 15 is: T r + 1 = C r 15 a x 1 / 3 15 - r - 1 b x 3 r For the coefficient of x - 15 in a x 1 / 3 - 1 b x 3 15 is: 5 - r 3 - 3 r = - 15 ⇒ 10 r