已知一非零的向量列{AN}满足；AN=(1.1),AN=(Xn,Yn)=1/2(Xn-1-Yn-1,Xn-1+Yn-1)(n>=2)(1)计算；IA1I,IA2I,IA3I;证明数列{IAnI}是等比数列

Question

已知一非零的向量列{AN}满足；AN=(1.1),AN=(Xn,Yn)=1/2(Xn-1 - Yn-1,Xn-1 +Yn-1)(n>=2)
(1)计算；IA1I,IA2I,IA3I;证明数列{IAnI}是等比数列

Answer 1

AN=(1.1),

x1=1,y1=1

AN=(Xn,Yn)=1/2(Xn-1 - Yn-1,Xn-1 +Yn-1)(n>=2)

∴xn=1/2[x(n-1)-y(n-1)]

   yn=1/2[x(n-1)+y(n-1)]

∴x2=1/2[1-1]=0,y2=1/2[1+1]= 1 ,A2=(0,1)

   x3=1/2[0-1]=-1/2,y3=1/2[0+1]=1/2,A3=(-1/2,1/2)

∴|A1|=√2,|A2|=1,|A3|=√2/2

n≥2时,

|An|=√[x²n+y²n] （上图）