（高中数列综合题）

∵数列{a[n]}点(a[n],a[n+1])在直线y=2x+1上
∴a[n+1]=2a[n]+1
即：a[n+1]+1=2(a[n]+1)
∵a[1]=1
∴{a[n]+1}是首项为a[1]+1=2,公比也是2的等比数列
即：a[n]+1=2*2^(n-1)=2^n
∴a[n]=2^n-1
∵数列{b[n]},b[n]/a[n]=1/a[1]+1/a[2]+1/a[3]+...+1/a[n-1] (n≥2,n∈N)
设c=1/a[1]+1/a[2]+1/a[3]+...+1/a[n-1]
∴b[n]=ca[n],b[n+1]=a[n+1](c+1/a[n])
∴b[n+1]a[n]-(b[n]+1)a[n+1]
=a[n+1](c+1/a[n])a[n]-(ca[n]+1)a[n+1]
=a[n+1](ca[n]+1)-a[n+1](ca[n]+1)
=0
(3)证明：
∵2^n-2^(n-1)=2^(n-1)≥1
∴2^n-1≥2^(n-1)
即：1/(2^n-1)≤1/2^(n-1)
∴1/a[1]+1/a[2]+1/a[3]+...+1/a[n-1]+1/a[n]
=1/(2^1-1)+1/(2^2-1)+1/(2^3-1)+...+1/(2^n-1)
≤1/2^0+1/2^1+1/2^2+...+1/2^(n-2)
=[1-1/2^(n-1)](1-1/2)
=2[1-1/2^(n-1)] (n≥2)