The coefficient of x^f in the expansion of 1 [3] (1-2 x)^2 is

Step-by-step solution

1 [3] (1-2 x)^2 = 1 (1-2 x)^ 2 / 3 =(1-2 x)^ -2 / 3 =1+ ( 2 3 )(2 x)+ 2 3 ( 2 3 +1 ) 2 ! (2 x)^2+ 2 3 ( 2 3 +1 ) ( 2 3 +2 ) 3 ! (2 x)^3+ + 2 3 ( 2 3 +1 ) ( 2 3 +2 ) ( 2 3 +(r-1) ) r ! (2 x)^r+ Here, the coefficient of x^r is : = 2 3 ( 2 3 +1 ) ( 2 3 +2 ) ( 2 3 +(r-1) ) r ! 2^r= 2 3 5 3 8 3 ( 3 r-1 3 ) r ! 2^r= 2 5 8 (3 r-1) r ! 3^r 2^r= 2 5 8 (3 r-1) r ! ( 2 3 )^r