Let the sum of the first n terms of the sequence an be Sn, A1 = 1, an = (Sn / N) + 2 (n-1) (n ∈ n *) and prove that the sequence an is an arithmetic sequence, Let the sum of the first n terms of a sequence an be Sn, A1 = 1, an = (Sn / N) + 2 (n-1) (n ∈ n *) 1. Prove: the sequence an is the arithmetic sequence, and write the expressions of an and Sn about n respectively 2. Is there a natural number n such that S1 + S2 / 2 + S3 / 3 + +Sn / N - (n-1) ^ 2 = 2013, if it exists, calculate the value of N, if it does not exist, please explain the reason

Let the sum of the first n terms of the sequence an be Sn, A1 = 1, an = (Sn / N) + 2 (n-1) (n ∈ n *) and prove that the sequence an is an arithmetic sequence, Let the sum of the first n terms of a sequence an be Sn, A1 = 1, an = (Sn / N) + 2 (n-1) (n ∈ n *) 1. Prove: the sequence an is the arithmetic sequence, and write the expressions of an and Sn about n respectively 2. Is there a natural number n such that S1 + S2 / 2 + S3 / 3 + +Sn / N - (n-1) ^ 2 = 2013, if it exists, calculate the value of N, if it does not exist, please explain the reason

When n ≥ 2, an = Sn / N + 2 (n-1) Sn = Nan - 2n (n-1) s (n-1) = (n-1) An-2 (n-1) (n-2) SN-S (n-1) = an = nan-2n (n-1) - (n-1) an + 2 (n-1) (n-2) an-a (n-1) = 4, which is the fixed value. Moreover, A1 = 1, and the sequence {an} is an arithmetic sequence with 1 as the first term and 4 as the tolerance. The general formula of an = 1 + 4 (n-1) = 4n-3 sequence {an} is an = 4n-3
That is Sn = [n (1 + 4n-3)] / 2
Sn / N = an - 2 (n-1) = 4n-3-2 (n-1) = 2n-1s1 / 1 + S2 / 2 +... + Sn / N - (n-1) & # 178; = 2 (1 + 2 +... + n) - N - (n-1) & # 178; = 2n (n + 1) / 2 - N - (n-1) & # 178; = 2N-1, let 2N-1 = 20112n = 2012
S1 = 1, that is S1 + S2 / 2 + S3 / 3 + +Sn/n—（n—1）^2=2013