Provide an appropriate response.
-What is the common ratio in the geometric sequence $\left. a _ { n } = \log \left[ 2 ^ { \left( 9 ^ { n } - 1 \right.} \right) \right]$ ?
A) 9
B) $\log 2$
C) $\log 9$
D) 2

Question

Quizplus · Accepted Answer

The common ratio of a geometric sequence can be found by dividing any term by the previous term. For the given sequence $a_n = \log[2^{(9^n - 1)}]$, we can find the ratio by examining the terms $a_{n+1}$ and $a_n$. The term $a_{n+1}$ would be $\log[2^{(9^{n+1} - 1)}]$. Simplifying the ratio $a_{n+1} / a_n$ involves properties of logarithms and exponents, leading to the realization that the base of the exponent increases by a factor of 9 for each successive term, hence the common ratio is 9.

Provide an Appropriate Response an=log⁡[2(9n−1)]\left. a _ { n } = \log \left[ 2 ^ { \left( 9 ^ { n } - 1 \right.} \right) \right]an​=log[2(9n−1)]

Provide an Appropriate Response $\left. a _ { n } = \log \left[ 2 ^ { \left( 9 ^ { n } - 1 \right.} \right) \right]$