1+i¹⁰+i²⁰+i³⁰ show that is real

Dear Student,

Please find below the solution to the asked query :

$$\begin{gathered} \mathrm{Let} \mathrm{z} = 1+{\mathrm{i}}^{10}+{\mathrm{i}}^{20}+{\mathrm{i}}^{30} \\ =1+{\left({\mathrm{i}}^2\right)}^5 + {\left({\mathrm{i}}^2\right)}^{10} + {\left({\mathrm{i}}^2\right)}^{15} \\ =1 + {\left(-1\right)}^5 + {\left(-1\right)}^{10} + {\left(-1\right)}^{15} \\ =1-1+1-1 \\ =0, \mathrm{which} \mathrm{is} \mathrm{a} \mathrm{real} \mathrm{number}. \\ \mathrm{So}, \mathrm{z} \mathrm{is} \mathrm{purely} \mathrm{real}. \end{gathered}$$

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