Simplify the following :
(i) `1+ i^(5)+i^(10)+i^(15)`
(ii) `(1+i)^(4)+(1+(1)/(i))^(4)`
(iii) `i^(n)+i^(n+1)+i^(n+2)+i^(n+3)`

Answer

N/a

Explanation

(i) `1+i^(5)+i^(10)+i^(15)`
`" "=1+i^(4)i+(i^(4))^(2)*i^(2)+(i^(4))^(3)*i^(3)`
`" "=1+(1)i+(1)^(2)(-1) + (1)^(3)(-i)`
`" "=1+i-1-i=0`.
(ii) `(1+i)^(4)+(1+(1)/(i))^(4)`
`" "=(1+i)^(4)+((i+1)/(i))^(4)`
`" "=(1+i)^(4)+((1+i)^(4))/(i^(4))`
`" "=(1+i)^(4)+(1+i)^(4)" "(becausei^(4)=1)`
`" "=2(1+i)^(4)=2[(1+i)^(2)]^(2)`
`" "=2(1+i^(2)+2i)^(2)=2(1-1+2i)^(2)`
`" "2(2i)^(2)=2(4i^(2))=8(-1)`
`" "=-8`.
(iii) `i^(n)+i^(n+1)+i^(n+2)+i^(n+3)`
`" "=i^(n)(1+i+i^(2)+i^(3))`
`" "=i^(n)(1+i-1-i)=i^(n)(0)`
= 0.