To find the fourth term of a geometric sequence where the first two terms are given as 27 and 9, we need to follow these steps:
1. Identify the first term (a1):
The first term of the sequence is [tex]\(a_1 = 27\)[/tex].
2. Identify the second term (a2):
The second term of the sequence is [tex]\(a_2 = 9\)[/tex].
3. Determine the common ratio (r):
A geometric sequence is defined by a common ratio, [tex]\(r\)[/tex], between consecutive terms. To find the common ratio, we divide the second term by the first term:
[tex]\[
r = \frac{a_2}{a_1} = \frac{9}{27} = \frac{1}{3}
\][/tex]
4. Use the formula for the nth term of a geometric sequence:
The formula to find the nth term of a geometric sequence is:
[tex]\[
a_n = a_1 \times r^{(n-1)}
\][/tex]
For the fourth term ([tex]\(a_4\)[/tex]), we plug in [tex]\(n = 4\)[/tex]:
[tex]\[
a_4 = a_1 \times r^{(4-1)}
\][/tex]
Simplify this to:
[tex]\[
a_4 = a_1 \times r^3
\][/tex]
5. Substitute the known values into the formula:
We know [tex]\(a_1 = 27\)[/tex] and [tex]\(r = \frac{1}{3}\)[/tex]. Plug these values into the formula:
[tex]\[
a_4 = 27 \times \left( \frac{1}{3} \right)^3
\][/tex]
Calculate the exponent and the product:
[tex]\[
a_4 = 27 \times \frac{1}{27} = 1
\][/tex]
Therefore, the fourth term of the geometric sequence is [tex]\(a_4 = 1\)[/tex].
