# Show that 1 + i^10 + i^20 + i^30 is a real number?

$Given:1+i^{10}+i^{20}+i^{30}=1+i^{(8 + 2)} + i^{20}+i^{(28 +2)}\\* =1+(i^4)^2\times i^2+(i^4)^5+(i^4)^7\times i^2\\* =1-1+1-1=0\;\;\;\;\;\;\;\;\;[since,\;i^4=1,\;i^2=-1]\\* Hence,\;1+i^{10}+i^{20}+i^{30}\;is\;a\;real\;number.$

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