Asked by Willard muliya
12,b,75 , are consecutive terms of geometric sequences.what it is common ratio for the sequences?
Answers
Answered by
GPT-4o mini
In a geometric sequence, each term is obtained by multiplying the previous term by a common ratio \( r \). Given the terms \( 12 \), \( b \), and \( 75 \), we can establish the following relationships:
1. The relationship between the first and second terms can be expressed as:
\[
b = 12r
\]
2. The relationship between the second and third terms can be expressed as:
\[
75 = br
\]
Substituting the expression for \( b \) from the first equation into the second equation gives us:
\[
75 = (12r)r
\]
or
\[
75 = 12r^2.
\]
To find \( r^2 \), we solve for \( r^2 \):
\[
r^2 = \frac{75}{12} = \frac{25}{4}.
\]
Taking the square root of both sides:
\[
r = \sqrt{\frac{25}{4}} = \frac{5}{2}.
\]
Thus, the common ratio \( r \) for the geometric sequence is:
\[
\boxed{\frac{5}{2}}.
\]
1. The relationship between the first and second terms can be expressed as:
\[
b = 12r
\]
2. The relationship between the second and third terms can be expressed as:
\[
75 = br
\]
Substituting the expression for \( b \) from the first equation into the second equation gives us:
\[
75 = (12r)r
\]
or
\[
75 = 12r^2.
\]
To find \( r^2 \), we solve for \( r^2 \):
\[
r^2 = \frac{75}{12} = \frac{25}{4}.
\]
Taking the square root of both sides:
\[
r = \sqrt{\frac{25}{4}} = \frac{5}{2}.
\]
Thus, the common ratio \( r \) for the geometric sequence is:
\[
\boxed{\frac{5}{2}}.
\]