数列an满足a1=1/3,an+1=an^2+an,记sn=1/(a1+1)+1/(a2+1)+.+1/(1+an),求s10的整数部分?

问题描述:

数列an满足a1=1/3,an+1=an^2+an,记sn=1/(a1+1)+1/(a2+1)+.+1/(1+an),求s10的整数部分?

∵a(n+1)=an(an+1)∴1/[a(n+1)]=1/an×1/(an+1)=1/an-1/(an+1)∴1/(an+1)=1/an-1/[a(n+1)]∴s10=1/a1-1/a2+1/a2-1/a3+……+1/a10-1/a11=1/a1-1/a11∵a1=1/3 a2=4/9 ……a4>1 a(n+1)>an∴a11>1 ∴0<1/a11<1∵...