Write this sum as:
[tex]i^{600}+i^{599}+i^{598}+\ldots+i+1=1+i+i^2+i^3+\ldots+i^{599}+i^{600}[/tex]
This is sum of a geometric series with [tex]n=601 [/tex] terms, first term [tex]a=1[/tex] and common ratio [tex]r=i[/tex]. So the sum:
[tex]S=a\dfrac{1-r^n}{1-r}=1\cdot\dfrac{1-i^{601}}{1-i}=\dfrac{1-i\cdot i^{600}}{1-i}=\dfrac{1-i\cdot(i^2)^{300}}{1-i}=\\\\\\=\dfrac{1-i\cdot(-1)^{300}}{1-i}=
\dfrac{1-i\cdot1}{1-i}=\dfrac{1-i}{1-i}=\boxed{1}[/tex]
