Get insightful responses to your questions quickly and easily on IDNLearn.com. Receive prompt and accurate responses to your questions from our community of knowledgeable professionals ready to assist you at any time.
Sagot :
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].
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].
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your questions are important to us at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.