Find the indicated term of the geometric sequence with the given description. The 3rd term is -1/3 and the 6th term is 9. Find the first and second terms.
To find the first and second terms of the geometric sequence, we first need to find the common ratio (r) of the sequence.
Given that the 3rd term is -1/3 and the 6th term is 9, we can use these two terms to find the common ratio.
Let the first term be a and the common ratio be r.
Using the formula for the nth term of a geometric sequence:
a_n = a * r^(n-1)
We know that:
a_3 = -1/3 = a * r^(3-1) = a * r^2
a_6 = 9 = a * r^(6-1) = a * r^5
We can set up a system of equations using these two equations:
-1/3 = a * r^2 ----(1)
9 = a * r^5 --------(2)
From equation (1), we can isolate a:
a = -1/(3r^2)
Substitute this into equation (2):
9 = (-1/(3r^2)) * r^5
9 = -r^3
r^3 = -9
r = -3
Now that we have found the common ratio, we can find the first term (a) using equation (1):
-1/3 = a * (-3)^2
-1/3 = 9a
a = -1/27
Therefore, the first term is -1/27 and the second term can be found by multiplying the first term by the common ratio:
a_2 = (-1/27) * (-3) = 1/9
So, the first term is -1/27 and the second term is 1/9.