题目内容
已知一列非零向量an满足:a1=(x1,y1),an=(xn,yn)=
(xn-1-yn-1,xn-1+yn-1)(n≥2).(1)证明{|an|}是等比数列;
(2)设θn=〈an-1,an〉,bn=2nθn-1,Sn=b1+b2+b3+…+bn,求Sn.
(1)证明:|an|=
=
=
|an-1|(n≥2).
∴
,且|a1|=
2≠0.∴{|an|}是公比为
的等比数列.
(2)解:∵an-1·an=(xn-1,yn-1)·
(xn-1-yn-1,xn-1+yn-1)=
(xn-12+yn-12)=
|an-1|2,
∴cos＜an,an-1＞=
.
∴θn=＜an,an-1＞=
.∴bn=2n·
-1=
π-1,
即Sn=(
-1)+( 
-1)+(
-1)+…+(
-1)= 
(1+2+…+n)-n
=
·
-n=
(n2+n)-n.
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