The first, third and fifth terms of a geometric sequence from arithmetic sequence. If the first term of the sequence is 3, find the 10th term of the geometric sequence

Answer 1

ar^2 - a = ar^4 - ar^2
but a = 3, so
3r^2 - 3 = 3r^4 - 3r^2
3r^4 - 6r^2 + 3 = 0
r^4 - 2r^2 + 1 = 0
(r^2 - 1)^2 = 0
r^2 = 1
r = ± 1

If r = 1, the terms of the GP would be 3, 3, 3, 3, ...
if r = -1, the terms of the GP would be 3, -3, 3, -3, ...

term 10 could be either 3, or -3

A sequence with a common ratio of 1 is technically a GP but is of
little interest, the same is true if r = -1, then the same terms would simple alternate between + and -1