Question

设数列{an}的前n项和为Sn,且(3-P)Sn+2*P(an)=P+3,其中P为常数,P

Answer 1

1.(3-p)sn+2p(sn-s(n-1))=p+3

(3+p)sn=2ps(n-1)+p+3

sn=2p/(p+3)s(n-1)+1

an+s(n-1)=2p/(p+3)s(n-1)+1

an=(p-3)/(p+3)s(n-1)+1

a(n-1)=(p-3)/(p+3)s(n-2)+1

an-a(n-1)=(p-3)/(p+3)(s(n-1)-s(n-2))

an-a(n-1)=(p-3)/(p+3)a(n-1)

an=2p/(p+3)a(n-1)

所以an为公比为2p/(p+3)的等比数列.

2.f(p)=2p/(p+3)

f(b(n-1))=2b(n-1)/(b(n-1)+3)

bn=3b(n-1)/(b(n-1)+3)

1/bn=1/3+1/b(n-1)

1/bn-1/b(n-1)=1/3

所以1/bn为等差数列,公差为1/3,首项为1/b1=1/a1,

1/bn=1/a1+(n-1)/3=(3+a1(n-1))/3*a1

bn=3a1/(3+a1(n-1))