Q.

If xn+iyn=(1+i)n then xn-1yn-xnyn-1=

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a

2n-1

b

2n+1

c

2n-2

d

22n-1

answer is D.

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Detailed Solution

xn+iyn=(1+i)n=2n 1+i2n xn+iyn=2ncosπ4+isinπ4n              = 2ncos4+isin4 xn=2n cos4,   yn=2n sin4  and xn-1=2n-1 cos(n-1)π4   yn-1=2n-1 sin(n-1)π4 xn-1 yn-xn yn-1=2n.2n-1sin4 cos(n-1)π4-cosπ4Tan(n-1)π4                                  =2n2 sin4-(n-1)π4=2n2.sinπ4                                  =2n2×12=2n2 =2n-1  

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If xn+iyn=(1+i)n then xn-1yn-xnyn-1=