aligned & 1 x(x+1)(x+2) (x+n) = A₀ x + A₁ x+1 & + A_n x+n , 0 i r A_r is equal to aligned

Step-by-step solution

aligned & Given, 1 x(x+1) (x+n) & = A₀ x + A₁ x+1 + + A_r x+r + + A_n x+n & 1 x(x+1) (x+r-1) (x+r+1) (x+n) & = x+r x A₀+ +A_r+ + x+r x+n A_n & _ x -r 1 x(x+1)(x+2) (x+n) & =0+ +0+A_r+ +0 & A_r= 1 -r(-r+1) (-1) 1 2 (-r+n) & aligned aligned & = (-1)^r r(r-1) (1)[1 2 (n-r)] A_r & = (-1)^r r !(n-r) ! aligned