Given that the point (1,1 / 3) is a point on the image of the function f (x) = a ^ x (a > 0 and a ≠ 1), the sum of the first n terms of the proportional sequence {an} is f (n) - C, and the sequence {BN} Given that the point (1,1 / 3) is a point on the function f (x) = a ^ X image, the sum of the first n terms of the proportional sequence an is f (x) - C, and the first term of the sequence BN is C, and the first n terms and Sn satisfy SN-S (n-1) = √ Sn √ s (n-1): ① the general term formula of the sequence an and BN; ② If the sum of the first n terms of the sequence {1 / BN * B (n 1)} is TN, ask TN From the meaning of the title 1）a=1/3,an=fn-c-（f（n-1）-c）=fn-f（n-1）=-2/3*（1／3） ^（n-1） The sum of the first n terms of an is (1 / 3) ^ n - 1 ∴c=1 And ∵ SN-S (n-1) = √ Sn + √ s (n-1) ∴√Sn-√Sn-1=1 ∴√Sn=n,Sn=n^2 ∴bn=Sn-Sn-1=2n-1 2) BN is substituted into 1 / bnbn BN + 1 = 1 / (2n-1) (2n + 1) = 1 / 2 (1 / 2n-1-1 / 2n + 1) ∴Tn=1/2（1-1/2n+1)=n/2n+1＞1000/2009 The solution is n > 1000 / 9 The minimum value of n is 112 ∴√Sn-√Sn-1=1 ∴√Sn=n,Sn=n^2 How is Sn = n deduced?

Given that the point (1,1 / 3) is a point on the image of the function f (x) = a ^ x (a > 0 and a ≠ 1), the sum of the first n terms of the proportional sequence {an} is f (n) - C, and the sequence {BN} Given that the point (1,1 / 3) is a point on the function f (x) = a ^ X image, the sum of the first n terms of the proportional sequence an is f (x) - C, and the first term of the sequence BN is C, and the first n terms and Sn satisfy SN-S (n-1) = √ Sn √ s (n-1): ① the general term formula of the sequence an and BN; ② If the sum of the first n terms of the sequence {1 / BN * B (n 1)} is TN, ask TN From the meaning of the title 1）a=1/3,an=fn-c-（f（n-1）-c）=fn-f（n-1）=-2/3*（1／3） ^（n-1） The sum of the first n terms of an is (1 / 3) ^ n - 1 ∴c=1 And ∵ SN-S (n-1) = √ Sn + √ s (n-1) ∴√Sn-√Sn-1=1 ∴√Sn=n,Sn=n^2 ∴bn=Sn-Sn-1=2n-1 2) BN is substituted into 1 / bnbn BN + 1 = 1 / (2n-1) (2n + 1) = 1 / 2 (1 / 2n-1-1 / 2n + 1) ∴Tn=1/2（1-1/2n+1)=n/2n+1＞1000/2009 The solution is n > 1000 / 9 The minimum value of n is 112 ∴√Sn-√Sn-1=1 ∴√Sn=n,Sn=n^2 How is Sn = n deduced?

√Sn-√S(n-1)=1
Let √ Sn = CN, the above formula can be reduced to cn-c (n-1) = 1,
This shows that the sequence {CN} is an arithmetic sequence, the tolerance is 1, and the first term is C1 = S1 = 1,
So CN = C1 + (n-1) × 1 = 1 + (n-1) × 1 = n,
That is √ Sn = n, so Sn = n ^ 2