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-
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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(an+1)/a(n+1)=bn/b(n+1);证明(1+1/a1)(1+1/a2)…(1+1/an)
证明(an+1)/a(n+1)=bn/b(n+1);证明(1+1/a1)(1+1/a2)…(1+1/an)
1=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
∴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
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