设数列a1,a2,…,an,满足a1=a2=1,a3=2,且对任何自然数n,都有anan+1an+2≠1,又anan+1an+2an+3=an+an+1+an+2+an+3,则a1+a2+…+a100的值是_________________.
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![设数列a1,a2,…,an,满足a1=a2=1,a3=2,且对任何自然数n,都有anan+1an+2≠1,又anan+1an+2an+3=an+an+1+an+2+an+3,则a1+a2+…+a100的值是_________________.](/uploads/image/z/11433403-19-3.jpg?t=%E8%AE%BE%E6%95%B0%E5%88%97a1%2Ca2%2C%E2%80%A6%2Can%2C%E6%BB%A1%E8%B6%B3a1%3Da2%3D1%2Ca3%3D2%2C%E4%B8%94%E5%AF%B9%E4%BB%BB%E4%BD%95%E8%87%AA%E7%84%B6%E6%95%B0n%2C%E9%83%BD%E6%9C%89anan%2B1an%2B2%E2%89%A01%2C%E5%8F%88anan%2B1an%2B2an%2B3%3Dan%2Ban%2B1%2Ban%2B2%2Ban%2B3%2C%E5%88%99a1%2Ba2%2B%E2%80%A6%2Ba100%E7%9A%84%E5%80%BC%E6%98%AF_________________.)
设数列a1,a2,…,an,满足a1=a2=1,a3=2,且对任何自然数n,都有anan+1an+2≠1,又anan+1an+2an+3=an+an+1+an+2+an+3,则a1+a2+…+a100的值是_________________.
设数列a1,a2,…,an,满足a1=a2=1,a3=2,且对任何自然数n,都有anan+1an+2≠1,又anan+1an+2an+3=an+an+1+an+2+an+3,则a1+a2+…+a100的值是_________________.
设数列a1,a2,…,an,满足a1=a2=1,a3=2,且对任何自然数n,都有anan+1an+2≠1,又anan+1an+2an+3=an+an+1+an+2+an+3,则a1+a2+…+a100的值是_________________.
看不懂你出的题- - !
anan+1an+2是个什么式子
an*a(n+1)*a(n+2)*a(n+3)=an+a(n+1)+a(n+2)+a(n+3)
a(n+1)*a(n+2)*a(n+3)*a(n+4)=a(n+1)+a(n+2)+a(n+3)+a(n+4)
两式相减,整理有
(a(n+4)-an)(a(n+1)*a(n+2)*a(n+3)-1)=0
由a1=1,a2=1,a3=2,求出a4=4,a2*a3*a4=8...
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an*a(n+1)*a(n+2)*a(n+3)=an+a(n+1)+a(n+2)+a(n+3)
a(n+1)*a(n+2)*a(n+3)*a(n+4)=a(n+1)+a(n+2)+a(n+3)+a(n+4)
两式相减,整理有
(a(n+4)-an)(a(n+1)*a(n+2)*a(n+3)-1)=0
由a1=1,a2=1,a3=2,求出a4=4,a2*a3*a4=8不等于1,所以a(n+1)*a(n+2)*a(n+3)-1不恒等于0
所以
a(n+4)-an=0
所以数列是a1=1,a2=1,a3=2,a4=4的循环数列
循环数列 1,1,2,4,1,1,2,4...
S100=(1+1+2+4)*25=200
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