Question

show that 1+i^10+i20+i^30is a real number.

Answer 1

i¹⁰=-1
i²⁰=1
i³⁰=-1
1+(-1)+1+(-1)=0
0 is a real number,
So, we can say that the value of 1+i¹⁰+i²⁰+i³⁰ is a real number.

Answer 2

We know that,
i = $\sqrt{-1}$


i¹⁰ = -1
i²⁰ = +1
i³⁰ = -1

Given L.H.S = 1 + i¹⁰ + i²⁰ + i³⁰ 
                    = 1 - 1 + 1 - 1
                    = 0
                    = Real number
                    = R.H.S