If 1.3+2.3^2+3.3^3+ .+ n .3^n= (2 n -1) 3^ a + b 3 then a and b are respectively

Step-by-step solution

1.3+2.3^2+3.3^3+ .+ n .3^ n = (2 n -1) 3^ a + b 4 Let us put 3= x . L.H.S: S=x+2 x^2+3 x^3+ .+n x^n (1) xs = x ^2+2 x ^3+3 x ^4+ .+ n x ^ n +1 (2) (1) aligned -(2) S - xS = ( x +2 x ^2+3 x ^3+ . & . .+ n x ^ n )- ( x ^2+2 x ^3 . & .+3 x ^4+ .+ n x ^ n +1 ) aligned S (1- x )= x + x ^2+ x ^3+ .+ x ^ n - nx ^ n +1 aligned & S (1- x )= x (1- x ^ n ) 1- x - nx x ^ n +1 & ~S = ( 1 x -1 ) ( - x ( x ^ n -1 )+ n x ^ n +1 ( x -1) x -1 ) aligned Put x=3 , S = 1 2 ( -3^ n +1 +3+2 n 3^ n +1 2 )= ( 3^ n +1 (2 n -1)+3 4 )