Write the value of
Question:

Write the value of $\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}$.

Solution:

$\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}$

$=\frac{i^{4 \times 148}+i^{4 \times 147+2}+i^{4 \times 147}+i^{146 \times 4+2}+i^{4 \times 146}}{i^{4 \times 145+2}+i^{4 \times 145}+i^{4 \times 144+2}+i^{4 \times 144}+i^{4 \times 143+2}}$

$=\frac{1+i^{2}+1+i^{2}+1}{i^{2}+1+i^{2}+1+i^{2}} \quad\left[\because i^{4}=1\right]$$=\frac{1-1+1-1+1}{-1+1-1+1-1} \quad\left[\because i^{2}=1\right]$

$=\frac{1}{-1}$

$=-1$

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