r = 3 and the 10th term of the geometric sequence is 19,683.
Let the common ratio of the geometric sequence be r.
The third term of the geometric sequence is 3r.
The fifth term of the geometric sequence is 3r^2.
Since the first, third, and fifth terms form an arithmetic sequence, we have:
3 + 3r = 2(3r^2)
3 + 3r = 6r^2
6r^2 - 3r - 3 = 0
r^2 - r/2 - 1 = 0
(r - 3)(r + 1/2) = 0
r = 3 or r = -1/2
If r = 3, then the 10th term of the geometric sequence is 3(3)^9 = 19,683
If r = -1/2, then the 10th term of the geometric sequence is 3(-1/2)^9 = -3/256
Since the first term of the sequence is 3, the common ratio must be positive. Therefore, r = 3 and the 10th term of the geometric sequence is 19,683.