Question
if {gn} is a geometric progressive with gn>0 for all 'n' natural number then prove that{ln(gn)} is Arthimetic progressive?
Answers
GPT-4o mini
To prove that the sequence \( \{ \ln(g_n) \} \) is an arithmetic progression when \( \{ g_n \} \) is a geometric progression, we start by defining the properties of the geometric progression.
1. Since \( \{ g_n \} \) is a geometric progression, we can express its terms as:
\[
g_n = g_1 r^{n-1}
\]
where \( g_1 > 0 \) is the first term and \( r > 0 \) is the common ratio.
2. Now, let's examine the natural logarithm of the terms of the sequence:
\[
\ln(g_n) = \ln(g_1 r^{n-1})
\]
3. Using the properties of logarithms, we can split this expression:
\[
\ln(g_n) = \ln(g_1) + \ln(r^{n-1}) = \ln(g_1) + (n-1) \ln(r)
\]
4. We can denote \( A = \ln(g_1) \) and \( d = \ln(r) \), and rewrite the distinct \( n \)-th term of the sequence:
\[
\ln(g_n) = A + (n-1)d
\]
5. Based on this expression, it is clear that \( \ln(g_n) \) can be expressed in the form of an arithmetic sequence:
\[
\ln(g_n) = A + (n-1)d
\]
6. The common difference \( d \) is constant (since it is just \( \ln(r) \)) and does not depend on \( n \). Thus, the sequence \( \{ \ln(g_n) \} \) is indeed an arithmetic progression.
### Conclusion:
We have shown that if \( \{ g_n \} \) is a geometric progression, then \( \{ \ln(g_n) \} \) can be expressed as a linear function of \( n \), confirming that \( \{ \ln(g_n) \} \) is an arithmetic progression. Hence, \( \{ \ln(g_n) \} \) is indeed an arithmetic progression.
1. Since \( \{ g_n \} \) is a geometric progression, we can express its terms as:
\[
g_n = g_1 r^{n-1}
\]
where \( g_1 > 0 \) is the first term and \( r > 0 \) is the common ratio.
2. Now, let's examine the natural logarithm of the terms of the sequence:
\[
\ln(g_n) = \ln(g_1 r^{n-1})
\]
3. Using the properties of logarithms, we can split this expression:
\[
\ln(g_n) = \ln(g_1) + \ln(r^{n-1}) = \ln(g_1) + (n-1) \ln(r)
\]
4. We can denote \( A = \ln(g_1) \) and \( d = \ln(r) \), and rewrite the distinct \( n \)-th term of the sequence:
\[
\ln(g_n) = A + (n-1)d
\]
5. Based on this expression, it is clear that \( \ln(g_n) \) can be expressed in the form of an arithmetic sequence:
\[
\ln(g_n) = A + (n-1)d
\]
6. The common difference \( d \) is constant (since it is just \( \ln(r) \)) and does not depend on \( n \). Thus, the sequence \( \{ \ln(g_n) \} \) is indeed an arithmetic progression.
### Conclusion:
We have shown that if \( \{ g_n \} \) is a geometric progression, then \( \{ \ln(g_n) \} \) can be expressed as a linear function of \( n \), confirming that \( \{ \ln(g_n) \} \) is an arithmetic progression. Hence, \( \{ \ln(g_n) \} \) is indeed an arithmetic progression.