Question
Find the 12th term of the geometric sequence 2, comma, minus, 10, comma, 50, comma, point, point, .2,−10,50,...
Answers
GPT 3.5
To find the 12th term of the geometric sequence, we first need to find the common ratio. The common ratio is found by dividing any term in the sequence by its previous term.
The common ratio of this sequence can be found by dividing each term by its previous term:
\[
\begin{align*}
\frac{2}{-10} &= -\frac{1}{5} \\
\frac{-10}{50} &= -\frac{1}{5} \\
\frac{50}{-0.2} &= -\frac{1}{5} \\
\end{align*}
\]
Since all of these ratios are equal to $-\frac{1}{5}$, we know that the common ratio is $-\frac{1}{5}$.
To find the 12th term, we can use the formula for the nth term of a geometric sequence:
\[ a_n = a_1 \cdot r^{(n-1)} \]
where $a_n$ is the nth term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the position of the term we want to find.
In this case, the first term $a_1$ is 2 and the common ratio $r$ is $-\frac{1}{5}$.
Substituting these values into the formula, we have:
\[ a_{12} = 2 \left(-\frac{1}{5}\right)^{(12-1)} \]
Simplifying, we have:
\[ a_{12} = 2 \left(-\frac{1}{5}\right)^{11} \]
This is the 12th term of the geometric sequence.
The common ratio of this sequence can be found by dividing each term by its previous term:
\[
\begin{align*}
\frac{2}{-10} &= -\frac{1}{5} \\
\frac{-10}{50} &= -\frac{1}{5} \\
\frac{50}{-0.2} &= -\frac{1}{5} \\
\end{align*}
\]
Since all of these ratios are equal to $-\frac{1}{5}$, we know that the common ratio is $-\frac{1}{5}$.
To find the 12th term, we can use the formula for the nth term of a geometric sequence:
\[ a_n = a_1 \cdot r^{(n-1)} \]
where $a_n$ is the nth term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the position of the term we want to find.
In this case, the first term $a_1$ is 2 and the common ratio $r$ is $-\frac{1}{5}$.
Substituting these values into the formula, we have:
\[ a_{12} = 2 \left(-\frac{1}{5}\right)^{(12-1)} \]
Simplifying, we have:
\[ a_{12} = 2 \left(-\frac{1}{5}\right)^{11} \]
This is the 12th term of the geometric sequence.