Question

Find the value of

$\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}-1$

Answer 1

Simplifying $\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}-1$.

$=\frac{{\left(i^2\right)}^{296}+{\left(i^2\right)}^{295}+{\left(i^2\right)}^{294}+{\left(i^2\right)}^{293}+{\left(i^2\right)}^{292}}{{\left(i^2\right)}^{291}+{\left(i^2\right)}^{290}+{\left(i^2\right)}^{289}+{\left(i^2\right)}^{288}+{\left(i^2\right)}^{287}}-1$

$=\frac{{\left(-1\right)}^{296}+{\left(-1\right)}^{295}+{\left(-1\right)}^{294}+{\left(-1\right)}^{293}+{\left(-1\right)}^{292}}{{\left(-1\right)}^{291}+{\left(-1\right)}^{290}+{\left(-1\right)}^{289}+{\left(-1\right)}^{288}+{\left(-1\right)}^{287}}-1$

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

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

$=-1-1$

$=-2$