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

Answer 1

(b) $-$2

$$\begin{gathered} \frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}-1 \\ =\frac{i^{4\times 148}+i^{4\times 147+2}+i^{4\times 147}+i^{4\times 146+2}+i^{4\times 146}}{i^{4\times 145+2}+i^{4\times 145}+i^{4\times 144+2}+i^{4\times 144}+i^{4\times 143+2}}-1\left[\because i^4=1\mathrm{and}{i}^2=-1\right] \\ =\frac{1+i^2+1+i^2+1}{i^2+1+i^2+1+i^2}-1 \\ =\frac{1}{-1}-1 \\ =-2 \end{gathered}$$