Question
find the common ratio of the geometric sequence minus, 16, comma, minus, 48, comma, minus, 144, comma, dot dot dot−16,−48,−144,...
Answers
Answered by
GPT-4o mini
In a geometric sequence, the common ratio \( r \) can be calculated using the formula:
\[
r = \frac{a_{n}}{a_{n-1}}
\]
where \( a_n \) is the nth term, and \( a_{n-1} \) is the previous term.
In the given sequence: \(-16, -48, -144, \ldots\)
Let's find the common ratio using the first two terms:
1. From the first term \(-16\) to the second term \(-48\):
\[
r = \frac{-48}{-16} = \frac{48}{16} = 3
\]
2. Now, let's confirm the common ratio using the second term \(-48\) and the third term \(-144\):
\[
r = \frac{-144}{-48} = \frac{144}{48} = 3
\]
Since both calculations give the same common ratio, we conclude that the common ratio \( r \) of the sequence is:
\[
\boxed{3}
\]
\[
r = \frac{a_{n}}{a_{n-1}}
\]
where \( a_n \) is the nth term, and \( a_{n-1} \) is the previous term.
In the given sequence: \(-16, -48, -144, \ldots\)
Let's find the common ratio using the first two terms:
1. From the first term \(-16\) to the second term \(-48\):
\[
r = \frac{-48}{-16} = \frac{48}{16} = 3
\]
2. Now, let's confirm the common ratio using the second term \(-48\) and the third term \(-144\):
\[
r = \frac{-144}{-48} = \frac{144}{48} = 3
\]
Since both calculations give the same common ratio, we conclude that the common ratio \( r \) of the sequence is:
\[
\boxed{3}
\]