For z C , if (1+z)^n=1+ ^n C₁ z+ ^n C₂ z^2+ ^n C_n z^n and _ r=0 ¹⁰⁰ 100 c_r( r x)= (2 x 2 )¹⁰⁰ k x , then k=

Step-by-step solution

It is given that for a complex number z , (1+z)^n=1+ ^n C₁ z+ ^n C₂ z^2+ ^n C₃ z^3+ + ^n C_n z^n Put z= x+i x Then, (1+ x+i x)^n=1+ ^n C₁( x+i x)+ ^n C₂( x+i x)^2+ + ^n C_n( x+i x)^n 2^n ( x 2 )^n ( x 2 +i x 2 )^n= [1+ ^n C₁ x+ ^n C₂ (2 x)+ ^n C₃ (3 x) . .+ + ^n C_n (n x) ]+i [ ^n C₁ x+ ^n C₂ (2 x) . .+ ^n C₃ (3 x)+ + ^n C_n (n x) ] 2^n ( x 2 )^n ( n x 2 +i n x 2 )= [1+ ^n C₁ x+ ^n C₂ (2 x)+ ^n C₃ (3 x) . .+ + ^n C_n (n x) ]+i [ ^n C₁ x+ ^n C₂ (2 x) . .+ ^n C₃ (3 x)+ + ^n C_n (n x) ] On comparing imaginary parts, w