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  4. Value of ( i 592+ i 590+ i 588+ i 586+ i 584/ i 582+ i 580+ i 578+ i 576+ i 574)-1 is

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Q. 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

Complex Numbers and Quadratic Equations

Solution:

Given expression
$=\frac{ i ^{10}\left( i ^{582}+ i ^{580}+ i ^{578}+ i ^{576}+ i ^{574}\right)}{ i ^{582}+ i ^{580}+ i ^{578}+ i ^{576}+ i ^{574}}-1$
$= i ^{10}-1=\left( i ^{2}\right)^{5}-1=(-1)^{5}-1=-1-1=-2$