设数列{an}满足a1=1/3,a(n+1)=an^2+an(n属于N*),记Sn=1/1+a1+1/1+a2+·····+1/1+an,则S10的整数部分为

Question

设数列{an}满足a1=1/3,a(n+1)=an^2+an(n属于N*),记Sn=1/1+a1+1/1+a2+·····+1/1+an,则S10的整数部分为

Answer 1

a(n+1)=an^2+an
a(n+1)=an(an+1)
1/a(n+1)=1/an(an+1)=[1/an-1/(an+1)]
1/(an+1)=1/an-1/(a(n+1)
Sn=1/1+a1+1/1+a2+·····+1/1+an,
=(1/a1-1/a2)+(1/a2-1/a3)+...+[1/an-1/a(n+1)]
=[1/a1-1/(a(n+1)]=3-1/a(n+1)
a1=1/3,a2=4/9,a3=52/81...an