數列級數

1.設< an >為等比級數,已知S3+S6 = 2S9
則公比 = ?

設公比為r
2(a+ar+ar^2)+ar^3+ar^4+ar^5=2(a+ar+ar^2+........ar^8)
ar^3+ar^4+ar^5=-2(ar^6+ar^7+ar^8)
ar^3(1+r+r^2)=-2ar^6(1+r+r^2)
ar^3不等於0，r^2+r+1 > 0，故r^3=-1/2，r=(-1/2)^(1/3)

2.設< an >之Sn = 2an - 1 , n = 1,2.....
< bn >滿足b1 = 3,bk+1 = ak + bk,k = 1,2.....
求< bn >的前n項和

Sn-Sn-1=an=2(an-an-1)
an=2an-1
故< an > 為首項1,公比2之等比數列
b2-b1=a1
b3-b2=a2
..............
bn-bn-1=an-1

bn-b1=2^(n-1) - 1，bn=2^(n-1)+2
b1+.................bn=2^n+2n-1

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