The third, fifth and seventh product of the increasing equal proportion sequence is 512. The three terms are subtracted by 1, 3 and 9 to form the equal difference sequence. Let Sn = A1 ^ 2 + A2 ^ 2 + +An ^ 2, find SN

The third, fifth and seventh product of the increasing equal proportion sequence is 512. The three terms are subtracted by 1, 3 and 9 to form the equal difference sequence. Let Sn = A1 ^ 2 + A2 ^ 2 + +An ^ 2, find SN

If the fifth term is x and the common ratio is Q, then
(x/q^2)*x*(x*q^2)=512
So x = 8 and the increasing common ratio is greater than 1
（8／q^2-1）+(8*q^2-9)=2*(8-3)
So q = root 2 or root (1 / 2) (rounding)
So the first term of the sequence is 2, q = root 2
An ^ 2 is a sequence with the first term of 4 and q = 2
So Sn = A1 ^ 2 + A2 ^ 2 + +an^2＝8*（2^n-1）.