JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Integral power of iota, Algebraic operations and Equality of complex numbers

  • question_answer
    If \[\sum\limits_{k=0}^{100}{{{i}^{k}}}=x+iy\], then the values of \[x\] and \[y\]are

    A) \[x=-1,y=0\]

    B)  \[x=1,y=1\]

    C) \[x=1,y=0\]

    D) \[x=0,y=1\]

    Correct Answer: C

    Solution :

    \[\sum\limits_{k=0}^{100}{{{i}^{k}}=x+iy,}\]Þ \[1+i+{{i}^{2}}\]\[+......+{{i}^{100}}=x+iy\] Given series is G.P. Þ  \[\frac{1.(1-{{i}^{101}})}{1-i}=x+iy\] Þ \[\frac{1-i}{1-i}=x+iy\] Þ \[1+0i=x+iy\] Equating real and imaginary parts, we get the required result.

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