Question

设数列an的前n项和Sn＝n2,数列bn满足bn＝an／an＋M(m∈整数)若b1,b2,b8成等

设数列an的前n项和Sn＝n2,数列bn满足bn＝an／an＋M(m∈整数)若b1,b2,b8成等比数列试求m的值

Answer 1

(1)n>=2
Sn=n^2
Sn-1=(n-1)^2
Sn-Sn-1=n^2-(n-1)^2=2n-1=an
(n>=2)
n=1,a1=S1=1,
a1=2-1=1
an=2n-1(n,N*)
bn=an/(an+m)=（2n-1)/(2n-1+m)
b1=1/(1+m),b2=3/(3+m),b8=15/(15+m)

b2^2=b1b8
9/(3+m)^2=1/(1+m)*15/(15+m)
3/(3+m）^2=5/(1+m)(15+m)
3(1+m)(15+m)=5(3+m)^2
3(15+16m+m^2)=5(9+6m+m^2)
45+48m+3m^2=45+30m+5m^2
2m^2-18m=0
m^2-9m=0
m=0,m=9
m,N*
m=9