The value of
\(
\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}
\)
is:
Show Hint
- -1
- 1
- 0
- $i$
The Correct Option is A
Solution and Explanation
Factor out the common lowest power of $i$ from the numerator and denominator.
Step 2: Analysis
Numerator $= i^{584}(i^{8} + i^{6} + i^{4} + i^{2} + 1)$. Denominator $= i^{574}(i^{8} + i^{6} + i^{4} + i^{2} + 1)$. The bracketed terms are identical and cancel out.
Step 3: Conclusion
Result $= i^{584} / i^{574} = i^{10} = (i^{4})^{2} \cdot i^{2} = 1^{2} \cdot (-1) = -1$.
Final Answer: (A)
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