Question:
The value of $\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}-1$ is
(a) −1
(b) −2
(c) −3
(d) −4
Solution:
(b) −2
$\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}-1$
$=\frac{i^{4 \times 148}+i^{4 \times 147+2}+i^{4 \times 147}+i^{4 \times 146+2}+i^{4 \times 16} 16}{i^{4} \times 15+2}+i^{\times 1 \times 45}+i^{4 \times 144+2}+i^{4 \times 144}+i^{4 \times 143+2}-1 \quad\left[\because i^{4}=1\right.$ and $\left.i^{2}=-1\right]$
$=\frac{1+i^{2}+1+i^{2}+1}{i^{2}+1+i^{2}+1+i^{2}}-1$
$=\frac{1}{-1}-1$
$=-2$