JEE Advanced2009MathematicsComplex NumberActual

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

Options

A1 2^
B1 3 2^
C1 2 2^
D1 4 2^

Correct answer

D. 1 4 2^

Step-by-step solution

Given that z= +i =e^ i aligned _ m=1 ¹⁵ Im (z^ 2 m-1 ) & = _ m=1 ¹⁵ Im (e^ i )^ 2 m-1 = _ m=1 ¹⁵ Im e^ i(2 m-1) & = + 3 + 5 + + 29 & = ( +29 2 ) ( 15 2 2 ) ( 2 2 ) = (15 ) (15 ) = 1 4 2^ aligned

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