A1 = 1, N, an, Sn are equal difference sequence. It is proved that {Sn + N + 2} is equal ratio sequence

A1 = 1, N, an, Sn are equal difference sequence. It is proved that {Sn + N + 2} is equal ratio sequence

Because n, an, Sn are arithmetic sequences
So 2An = Sn + n
And because an = sn-sn-1
So Sn + n = 2sn-1 + 2n
Add 2 Sn + N + 2 = 2sn-1 + 2n + 2 on both sides at the same time
On the right, Sn + N + 2 = 2sn-1 + 2n + 2-2 + 2
That is Sn + N + 2 = 2sn-1 + 2 (n-1) + 4
That is, Sn + N + 2 = 2 [sn-1 + (n-1) + 2]
So {Sn + N + 2} is an equal ratio sequence with a common ratio of 2

It is known that each item of the equal ratio sequence an is a real number, and the common ratio is Q, the sum of the first n items is Sn, and S3, S6 and S9 are equal difference sequences. (1) find the value of Q; (2) prove that A2, a8 and A5 are equal difference sequences

(1) Let the common ratio of the equal ratio sequence {an} be qs3, S6 and S9, then 2s6 = S3 + S9. When q = 1, Sn = Na1  12a1 = 3A1 + 9a1, which is consistent with the meaning of the problem. When Q ≠ 1, then 2A1 (Q ^ 6-1) / (Q-1) = A1 (Q ^ 3-1) / (Q-1) + A1 (Q ^ 9-1) / (Q-1) 2q ^ 6-2 = q ^ 3-1 + Q ^ 9-1q ^ 9-2q ^ 6 + Q ^ 3 = 0, about Q ^ 3Q ^ 6-2q