数列{an}满足a1=a2=1,an+an+1+an+2=cos2nπ/3若数列{an}的前n项和为sn则s2012的值
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数列{an}满足a1=a2=1,an+an+1+an+2=cos2nπ/3若数列{an}的前n项和为sn则s2012的值
![数列{an}满足a1=a2=1,an+an+1+an+2=cos2nπ/3若数列{an}的前n项和为sn则s2012的值](/uploads/image/z/17849915-35-5.jpg?t=%E6%95%B0%E5%88%97%7Ban%7D%E6%BB%A1%E8%B6%B3a1%3Da2%3D1%2Can%2Ban%2B1%2Ban%2B2%3Dcos2n%CF%80%2F3%E8%8B%A5%E6%95%B0%E5%88%97%7Ban%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%E4%B8%BAsn%E5%88%99s2012%E7%9A%84%E5%80%BC)
∵2010=670×3
∴a2010+a2011+a2012=cos(2010×2π/3)=cos(670×2π)=cos(2π)=1
同理:
a2007+a2008+a2009=cos(2007×2π/3)=cos(2π)=1
.
a3+a4+a5=cos(2π)=1
∴s2012=(a2012+a2011+a2010)+(a2009+a2008+a2007)+.+(a5+a4+a3)+a2+a1
=670×1+a2+a1
=670+2
=672
∴a2010+a2011+a2012=cos(2010×2π/3)=cos(670×2π)=cos(2π)=1
同理:
a2007+a2008+a2009=cos(2007×2π/3)=cos(2π)=1
.
a3+a4+a5=cos(2π)=1
∴s2012=(a2012+a2011+a2010)+(a2009+a2008+a2007)+.+(a5+a4+a3)+a2+a1
=670×1+a2+a1
=670+2
=672
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