Let an be an arithmetic sequence with nonzero tolerance, Sn be the sum of the first n terms of an, and S3 ^ 2 = 9s2, S4 = 4s2, The formula of finding sequence an

(a1 + a2 + a3)^2 = 9 (a1 + a2)
(a1 + a2 + a3 + a4) = 4(a1 + a2)
If the tolerance is D, then
(a2 - d + a2 + a2 + d)^2 = 9(a2 - d + a2)
(a2 -d + a2 + a2+d + a2 + 2d) = 4(a2 - d + a2)
9a2^2 = 9 (2a2 - d)
4a2 + 2d = 4(2a2 -d)
a2^2 = 2a2 -d
2a2 = 3d
a^2 = 2a2 - 2a2 /3
a^2 = 4a2 /3
A2 = 0 or 4 / 3
When A2 = 0, d = 0, the whole sequence is a 0 sequence, rounding off
When A2 = 4 / 3, d = 8 / 9
a1 = a2 - d = 4/9
an = a1 + (n-1)d = 4/9 + 8(n-1)/9 = 4(2n-1)/9
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Appendix inspection:
a1 = 4/9
a2 = 12/9
a3 = 20/9
a4 = 28/9
S2 = 16/9
S4 = 64/9
S4 = 4s2
S3 = 36/9 = 4
S3^2 = 16
9S2 = 16
S3 ^ 2 = 9s2
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