To address this problem, let's use the properties of a geometric sequence. In a geometric sequence, each term after the first is the product of the previous term and a constant ratio, denoted as [tex]\( r \)[/tex].
Given:
- The first term [tex]\( a \)[/tex] is 27.
- The third mean (or the third term in the sequence) is 1.
1. Identify the Ratio:
To find the common ratio [tex]\( r \)[/tex], we can use the fact that the third term [tex]\( g_3 \)[/tex] is given by the formula for a geometric sequence:
[tex]\[
g_3 = a \cdot r^{3-1}
\][/tex]
Substituting the known values:
[tex]\[
1 = 27 \cdot r^2
\][/tex]
Solving for [tex]\( r^2 \)[/tex]:
[tex]\[
r^2 = \frac{1}{27}
\][/tex]
Taking the square root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[
r = \sqrt{\frac{1}{27}} \approx 0.19245008972987526
\][/tex]
2. Find the Second Term (First Mean):
The first mean (second term of the sequence) [tex]\( g_2 \)[/tex] is given by:
[tex]\[
g_2 = a \cdot r
\][/tex]
Substituting the known values:
[tex]\[
g_2 = 27 \cdot 0.19245008972987526 \approx 5.196152422706632
\][/tex]
3. Conclusion:
The common ratio [tex]\( r \)[/tex] is approximately [tex]\( 0.19245008972987526 \)[/tex] and the first mean (second term [tex]\( g_2 \)[/tex]) is approximately [tex]\( 5.196152422706632 \)[/tex].
Therefore, we have found the required information about the geometric sequence, determining both the ratio and the number of mean terms. The given context was calculating the second term [tex]\( g_2 \)[/tex] and identifying the ratio [tex]\( r \)[/tex].
