Q110 / 180 · COMEDK 2024 (Afternoon Shift) · Mathematics

$\text { The value of } \dfrac{i^{1004}+i^{1006}+i^{1008}+i^{1010}+i^{1012}}{i^{510}+i^{508}+i^{506}+i^{504}+i^{502}} \text { is }$

  • A
    1
  • B
    $-$i
  • C
    i
  • Correct answer
    D
    $-1$

Solution

The powers of $i$ repeat in a cycle of $4$: $i^{1} = i$, $i^{2} = -1$, $i^{3} = -i$, $i^{4} = 1$.

For the numerator, we have $i^{1004} + i^{1006} + i^{1008} + i^{1010} + i^{1012}$.
Since $1004, 1008, 1012$ are multiples of $4$, $i^{1004} = i^{1008} = i^{1012} = 1$.
Since $1006 = 1004 + 2$, $i^{1006} = i^{2} = -1$.
Since $1010 = 1008 + 2$, $i^{1010} = i^{2} = -1$.
Numerator $= 1 + (-1) + 1 + (-1) + 1 = 1$.

For the denominator, we have $i^{510} + i^{508} + i^{506} + i^{504} + i^{502}$.
Since $508, 504$ are multiples of $4$, $i^{508} = i^{504} = 1$.
Since $510 = 508 + 2$, $i^{510} = i^{2} = -1$.
Since $506 = 504 + 2$, $i^{506} = i^{2} = -1$.
Since $502 = 504 - 2$, $i^{502} = i^{-2} = \dfrac{1}{i^{2}} = \dfrac{1}{-1} = -1$.
Denominator $= (-1) + 1 + (-1) + 1 + (-1) = -1$.

The value of the expression is $\dfrac{1}{-1} = -1$.

Answer: $-1$

Hence, the correct answer is option: 'D'.

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