已知数列an=2-3n,求1╱（an×an+1）的 - 知识问答

已知数列an=2-3n,求1╱（an×an+1）的前n项和sn

1个回答

裂项求和
an=2-3n
∴ a(n+1)-a(n)=-3
∴ 1/(an*a(n+1))
=(-1/3)*(-3)/(an*a(n+1))
=(-1/3)*[a(n+1)-a(n)]/(an*a(n+1))
=(-1/3)*[1/a(n)-1/a(n+1)]
∴ Sn=(-1/3)[1/a1-1/a2+1/a2-1/a3+.+1/a(n)-1/a(n+1)]
=(-1/3)[1/a1-1/a(n+1)]
=-(1/3)*[1/(-1)-1/(2-3n-3)]
=(-1/3)*[-1+1/(3n+1)]
=(1/3)[1-1/(3n+1)]
=(1/3)*3n/(3n+1)
=n/(3n+1)

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