It's getting complex 2

Algebra Level 2

i 587 + i 629 + i 777 + i 847 + i 995 i 577 + i 619 + i 767 + i 837 + i 985 = ? \large \dfrac{i^{587}+i^{629}+i^{777}+i^{847}+i^{995}}{i^{577}+i^{619}+i^{767}+i^{837}+i^{985}}= \ ? \

Clarification : i = − 1 i=\sqrt{-1} .

− 1 -1 1 1 − i -i i i

Rohit Udaiwal by Rohit Udaiwal , 20, India

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2 solutions

Chew-Seong Cheong
Dec 16, 2015

i 587 + i 629 + i 777 + i 847 + i 995 i 577 + i 619 + i 767 + i 837 + i 985 = i 10 ( i 577 + i 619 + i 767 + i 837 + i 985 ) i 577 + i 619 + i 767 + i 837 + i 985 = i 10 = ( i 4 ) 2 ( i 2 ) = 1 2 ( − 1 ) = − 1 \begin{aligned} \frac{i^{587}+i^{629}+i^{777}+i^{847}+i^{995}}{i^{577}+i^{619}+i^{767}+i^{837}+i^{985}} & = \frac{i^{10}\left(i^{577}+i^{619}+i^{767}+i^{837}+i^{985}\right)}{i^{577}+i^{619}+i^{767}+i^{837}+i^{985}} \\ & = i^{10} = \left(i^4\right)^2 \left(i^2\right) = 1^2(-1) = \boxed{-1} \end{aligned}

8 Helpful 0 Interesting 1 Brilliant 0 Confused
Aareyan Manzoor
Dec 15, 2015

write as i 10 ( 5 t e r m s ) 5 t e r m s \dfrac{i^{10}(5 terms)}{5terms} since ther are five terms, their sum will be non-zero( since i n = 1 , − 1 , i , − i i^n=1,-1,i,-i 5 terms will not allow to cancel). after cancellation i 10 = i 2 = − 1 i^{10}=i^2=\boxed{-1}

6 Helpful 0 Interesting 0 Brilliant 0 Confused

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