数列bn满足b1=1,b(n+1)=2bn+1,若数列an满足a1=1,an=bn[1/b1+1/b2+…+1/b(n-1)],n≥2且n为正整数,

题目

数列bn满足b1=1,b(n+1)=2bn+1,若数列an满足a1=1,an=bn[1/b1+1/b2+…+1/b(n-1)],n≥2且n为正整数,
数列bn满足b1=1,b(n+1)=2bn+1,若数列an满足a1=1,an=bn[1/b1+1/b2+…+1/b(n-1)],n≥2且n为正整数,证明(1+1/a1)(1+1/a2)…(1+1/an)

答案

b1=1,b=2bn+1,∴b+1=2(bn+1),∴bn+1=(b1+1)*2^(n-1)=2^n,∴bn=2^n-1.a=b(1/b1+1/b2+……+1/bn),∴a/b=1/b1+1/b2+……+1/bn=an/bn+1/bn=(an+1)/bn,∴（an+1)/a=bn/b.n>=4时2^n>=n^,1/bn