The value of _ x 1 [ _ K=1 ¹⁰⁰-x^K ]-100 x-1 is

Step-by-step solution

_ x 1 [ _ K=1 ¹⁰⁰ x^K ]-100 (x-1) aligned & = _ x 1 (x+x^2+x^3+ +x¹⁰⁰ )-100 (x-1) & = _ x 1 (x-1)+ (x^2-1 )+ (x^3-1 )+ + (x¹⁰⁰-1 ) (x-1) aligned aligned = _ x 1 ( x-1 x-1 )+ ( x^2-1 x-1 )+ ( x^3-1 x-1 ) . & & .+ + ( x¹⁰⁰-1 x-1 ) aligned aligned = _ x 1 ( x-1 x-1 )+ _ x 1 ( x^2-1 x-1 ) & + _ x 1 ( x^3-1 x-1 ) & + + _ x 1 ( x¹⁰⁰-1 x-1 ) aligned aligned & =1+2+3+ +100 & = 100= 100 (100+1) 2 aligned =50 101=5050 n= n(n+1) 2 .