Question

Find the fourth term of the geometric sequence 2​, negative 6​, 18​, ...
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Part 1
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Answer 1

To find the fourth term of the geometric sequence, we first need to determine the common ratio between the terms.

The first few terms of the sequence are:
- \( a_1 = 2 \)
- \( a_2 = -6 \)
- \( a_3 = 18 \)

The common ratio (\( r \)) can be found by dividing the second term by the first term:

\[
r = \frac{a_2}{a_1} = \frac{-6}{2} = -3
\]

We can verify the common ratio with the next term:

\[
r = \frac{a_3}{a_2} = \frac{18}{-6} = -3
\]

Now that we know the common ratio is \(-3\), we can find the fourth term (\( a_4 \)) using the formula of a geometric sequence:
\[
a_n = a_1 \cdot r^{(n-1)}
\]
For the fourth term (\( a_4 \)):
\[
a_4 = a_1 \cdot r^{(4-1)} = 2 \cdot (-3)^3
\]

Calculating \( (-3)^3 \):
\[
(-3)^3 = -27
\]
Now plug this back into the equation for \( a_4 \):
\[
a_4 = 2 \cdot (-27) = -54
\]

Thus, the fourth term of the sequence is:
\[
\boxed{-54}
\]