Certainly! Let's walk through the steps to find the number of geometric means [tex]\( n \)[/tex] between [tex]\( \frac{1}{25} \)[/tex] and 25, given that the ratio of the first geometric mean ([tex]\( G_1 \)[/tex]) to the fifth geometric mean ([tex]\( G_5 \)[/tex]) is [tex]\( 1:25 \)[/tex].
### Step-by-Step Solution:
1. Identify the values:
- The first term ([tex]\( A \)[/tex]) is [tex]\( \frac{1}{25} \)[/tex].
- The last term ([tex]\( B \)[/tex]) is 25.
2. General Formula for the terms of a geometric sequence:
- Let the ratio between consecutive terms be [tex]\( r \)[/tex].
- Therefore, the sequence would be:
[tex]\[
\frac{1}{25}, \ G_1, \ G_2, \ \ldots, \ G_n, \ 25
\][/tex]
- The terms in this sequence can be expressed as:
[tex]\[
G_1 = \left(\frac{1}{25}\right) r, \quad G_2 = \left(\frac{1}{25}\right) r^2, \quad \ldots, \quad G_n = \left(\frac{1}{25}\right) r^n
\][/tex]
3. Given information:
- The ratio of the first geometric mean to the fifth geometric mean is given by:
[tex]\[
\frac{G_1}{G_5} = \frac{\left(\frac{1}{25}\right) r}{\left(\frac{1}{25}\right) r^5} = \frac{1}{r^4} = \frac{1}{25}
\][/tex]
4. Solve for [tex]\( r \)[/tex]:
- From the equation:
[tex]\[
\frac{1}{r^4} = \frac{1}{25}
\][/tex]
We obtain:
[tex]\[
r^4 = 25
\][/tex]
- Taking the fourth root of both sides, we find:
[tex]\[
r = \sqrt[4]{25} = \left(25\right)^{\frac{1}{4}} \approx 2.23606797749979
\][/tex]
5. Find [tex]\( n \)[/tex]:
- The sequence ends with the term [tex]\( 25 \)[/tex], which means:
[tex]\[
25 = \left(\frac{1}{25}\right) r^{n+1}
\][/tex]
- Substituting the value of [tex]\( r \)[/tex]:
[tex]\[
25 = \left(\frac{1}{25}\right) \left(2.23606797749979\right)^{n+1}
\][/tex]
- Simplify:
[tex]\[
25 = \frac{\left(2.23606797749979\right)^{n+1}}{25}
\][/tex]
- Multiply both sides by 25:
[tex]\[
625 = \left(2.23606797749979\right)^{n+1}
\][/tex]
- Take the logarithm of both sides (base 10 or natural logarithm):
[tex]\[
\log(625) = (n+1) \log(2.23606797749979)
\][/tex]
- Solve for [tex]\( n+1 \)[/tex]:
[tex]\[
n+1 = \frac{\log(625)}{\log(2.23606797749979)} \approx 8
\][/tex]
- Hence, [tex]\( n \)[/tex] is:
[tex]\[
n = 8 - 1 \approx 7
\][/tex]
Therefore, the number of geometric means [tex]\( n \)[/tex] between [tex]\( \frac{1}{25} \)[/tex] and 25 is approximately 7.
