Let a sequence x1,x2,x3,... of complex number be defined by x1=0,xn+1=x2n−i for n>1 where i2=−1. Find the distance of x2000 from x1997 in the complex plane. If the distance is k, then find k2.
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Solution
x1=0 x2=02−i=−i x3=(−i)2−i=−1−i=−(1+i) x4=[−(1+i)]2−i=2i−i=i x5=(i)2−i=−1−i=x3 ∴x6=x4 and hence x7=x5 and so on, ∴x2n=i, x2n+1=−(1+i) ∴x2000=i=(0,1) and x1997=−1−i=(−1,−1) in the complex plane. So, the distance between x2000 and x1997 is √1+4=√5.