To find the common ratio in a geometric sequence, we can use the information provided regarding the second and fourth terms.
1. Define the given terms:
- The second term ( [tex]\(A_2\)[/tex] ) of the geometric sequence is 20.
- The fourth term ( [tex]\(A_4\)[/tex] ) of the geometric sequence is 11.25.
2. Use the general formula for the [tex]\(n\)[/tex]th term in a geometric sequence:
[tex]\[
A_n = A_1 \cdot r^{(n-1)}
\][/tex]
Where:
- [tex]\(A_n\)[/tex] is the [tex]\(n\)[/tex]th term,
- [tex]\(A_1\)[/tex] is the first term,
- [tex]\(r\)[/tex] is the common ratio,
- [tex]\(n\)[/tex] is the term number.
3. Write the equations for the given terms:
- For [tex]\(A_2\)[/tex]:
[tex]\[
A_2 = A_1 \cdot r^1
\][/tex]
Given [tex]\(A_2 = 20\)[/tex], we get:
[tex]\[
20 = A_1 \cdot r
\][/tex]
- For [tex]\(A_4\)[/tex]:
[tex]\[
A_4 = A_1 \cdot r^3
\][/tex]
Given [tex]\(A_4 = 11.25\)[/tex], we get:
[tex]\[
11.25 = A_1 \cdot r^3
\][/tex]
4. Divide the second equation by the first equation to eliminate [tex]\(A_1\)[/tex]:
[tex]\[
\frac{A_1 \cdot r^3}{A_1 \cdot r} = \frac{11.25}{20}
\][/tex]
Simplifies to:
[tex]\[
r^2 = \frac{11.25}{20}
\][/tex]
5. Calculate [tex]\(r^2\)[/tex]:
[tex]\[
r^2 = \frac{45}{4} \div 20 = \frac{45}{80} = 0.5625
\][/tex]
6. Find [tex]\(r\)[/tex] by taking the square root of [tex]\(r^2\)[/tex]:
[tex]\[
r = \sqrt{0.5625} = 0.75
\][/tex]
Therefore, the common ratio [tex]\(r\)[/tex] in this geometric sequence is 0.75.
