Let z= +i . Then, the value of _ m=1 ¹⁵ I_m (z^ 2 m-1 ) at =2^ is

Step-by-step solution

z= +i =e^ i [in Euler's form] Using Demoivre theorem aligned z^2 & = 2 +i 2 z^3 & = 3 +i 3 aligned Now, according to the question gathered _ m=1 ¹⁵ I_m (z^ 2 m-1 ) Imaginary part of (z+z^3+z^5+ +z²⁹ ) + 3 + 5 + 29 ) 15 (2 ) 2 2 2 [ ( +(15-1) 2 2 ) ] [ + ( + )+ to n term n 2 2 [ ( +(n-1) 2 ) ] gathered aligned & Here, = , =2 , n=15] & 15 (15 ) aligned Now, =2^ aligned & = 30^ 30^ 2^ = 1 2 1 2 1 2^ & = 1 4 2^ aligned