If f_n(x)= 1 2 n [ ^ 2 n x+ ^ 2 n x ] , then f₁(x)+f₂(x)-f₃ (x) =

Step-by-step solution

aligned & f_n(x)= 1 2 n [ ^ 2 n x+ ^ 2 n x ] & f₁(x)+f₂(x)+f₃(x) & = 1 2 [ ^2 x+ ^2 x ]+ 1 4 [ ^4 x+ ^4 x ]- 1 6 [ ^6 x+ ^6 x ] & = 1 2 1+ 1 4 [ ^4 x+ ^4 x ]- 1 6 [ ( ^2 x+ ^2 x ) ( ^4 x+ ^4 x- ^2 x ^2 x ) ] & = 1 2 + 1 4 [ ^4 x+ ^4 x ]- 1 6 [ ^4 x+ ^4 x- ^2 x ^2 x ] & = 1 2 + 1 12 [ ^4+ ^4 x ]+ 1 6 ^2 x ^2 x & = 1 2 + 1 12 [ ( ^2 x+ ^2 x )^2-2 ^2 x ^2 x ]+ 1 6 ^2 x ^2 x & = 1 2 + 1 12 1- 1 6 ^2 x ^2 x+ 1 6 ^2 x ^2 x & = 1 2 + 1 12 = 6+1 12 = 7 12 aligned