If n is an integer which leaves remainder one when divided by three, then (1+3i)n +(1−3i)n =

Given n=3m+1  where m∈Z (1+3i)n+(1-3i)n=2n[(12)+(3/2)i]n+2n[(12)−(3/2)i]n=2n[cos(π/3)+isin(π/3)]n+2n[cos(π/3)−isin(π/3)]"                    by Demovier's theorem=2n[cos(nπ/3)+isin(nπ/3)]+[cos(nπ/3)−isin(nπ/3)]=2n2 cosnπ3=2n+1cos(nπ/3)    =2n+1cos(mπ+π/3)=2n+1(−1)m1/2=(−1)m2n=(−1)3m2n=--13m+12n=−(−2)n.    ( since n=3m+1)