Given the arithmetic sequence [an], Sn = [(an + 1) / 2] ^ 2, find the general term formula of an N is subscript, the result of my own calculation is extremely troublesome.

Given the arithmetic sequence [an], Sn = [(an + 1) / 2] ^ 2, find the general term formula of an N is subscript, the result of my own calculation is extremely troublesome.

∵ arithmetic sequence {a [n]}, s [n] = [(a [n] + 1) / 2] ^ 2
∴4S[n]=a[n]^2+2a[n]+1
∵4S[n+1]=a[n+1]^2+2a[n+1]+1
By subtracting the above two expressions, we get:
4a[n+1]=a[n+1]^2-a[n]^2+2a[n+1]-2a[n]
2(a[n+1]+a[n])=(a[n+1]+a[n])(a[n+1]-a[n])
If a [n + 1] + a [n] = 0, that is: a [n + 1] = - a [n]
∵a[1]=S[1]=[(a[1]+1)/2]^2
∴a[1]=1
{a [n]} is an equal ratio sequence with the first term of 1 and the common ratio of - 1
This is contrary to the condition that {a [n]} is an arithmetic sequence
∴a[n+1]+a[n]≠0
A [n + 1] - a [n] = 2, that is, the tolerance is 2
∵a[1]=1
∴a[n]=1+2(n-1)=2n-1