Question

Find the values of the following expressions :

$\left(i\right)i^{49}+i^{68}+i^{89}+i^{110}\left(ii\right)i^{30}+i^{80}+i^{120}\left(iii\right)i+i^2+i^3+i^4\left(iv\right)i^5+i^{10}+i^{15}\left(v\right)\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}\left(vi\right)1+i^2+i^4+i^6+i^8+...+i^{20}$

Answer 1

$\left(i\right)i^{49}+i^{68}+i^{89}+i^{110}=i^{4+12}\times i^1+i^{4\times 17}+i^{4\times 22}+i^1+i^{4\times 27}\times i^2=i+1+i-1=2i\therefore i^{49}+i^{68}+i^{89}+i^{110}=2i\left(ii\right)i^{30}+i^{80}+i^{120}=i^{4\times 7}+i^2+i^{4\times 20}+i^{4\times 30}=1\times i^2+1+1=-1+1+1=1\therefore i^{30}+i^{80}+i^{120}=1\left(iii\right)i+i^2+i^3+i^4=1+(-1)+(-i)+1=0\therefore i+i^2+i^3+i^4=0\left(iv\right)i^5+i^{10}+i^{15}=i^{4\times 1}\times i^1+i^{4\times 2}\times i^2+i^{4\times 3}\times i^3=1\times i+1\times i^2\times 1\times i^3=i-1-i=-1\therefore i^4+i^{10}+i^{15}=-1\left(v\right)\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}\frac{i^{4\times 148}+i^{148}\times i^2+i^{4\times 147}+i^{4\times 146}\times i^2\times i^{4\times 146}}{i^{4\times 145}+i^2+i^{4\times 145}+i^{4\times 144}+i^2+i^{4\times 144}+i^{4\times 143}\times i^2}=\frac{1+1\times i^2+1+1\times i^2+1}{1\times i^2+1+1\times i^2+1+1\times i^2}=\frac{1-1+1-1+1}{-1+1-1+1-1}=\frac{1}{-1}=-1\therefore \frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}=-1\left(vi\right)1+i^2+i^4+i^6+i^8+....+i^{20}=1+i^2+i^4+i^{4\times 1}\times i^2+i^{4\times 2}\times i^2+i^{4\times 3}\times i^2+i^{4\times 4}+i^{4\times 4}\times i^2+i^{4\times 5}=1-1+1+1\times i^2+1+1\times i^2+1+1\times i^2+1+1\times i^2+1=1-1+1-1+1-1+1-1+1-1+2=1$