Question

geometric sequence is -4,-12,-36
1) create an equation for the situation
2) find the 12th term of the sequence

Answer 1

To analyze the geometric sequence given: \(-4, -12, -36\), we can identify the first term and the common ratio.

### 1) Create an equation for the situation

The first term (\(a\)) of the sequence is:

\[ a = -4 \]

To find the common ratio (\(r\)), we can divide the second term by the first term:

\[
r = \frac{-12}{-4} = 3
\]

Now we can express the \(n\)-th term of a geometric sequence as:

\[
a_n = a \cdot r^{(n-1)}
\]

Substituting \(a = -4\) and \(r = 3\):

\[
a_n = -4 \cdot 3^{(n-1)}
\]

### 2) Find the 12th term of the sequence

To find the 12th term (\(a_{12}\)), we can substitute \(n = 12\) into the equation:

\[
a_{12} = -4 \cdot 3^{(12-1)} = -4 \cdot 3^{11}
\]

Calculating \(3^{11}\):

\[
3^{11} = 177147
\]

Now we can calculate \(a_{12}\):

\[
a_{12} = -4 \cdot 177147 = -708588
\]

### Conclusion

1) The equation for the geometric sequence is:

\[
a_n = -4 \cdot 3^{(n-1)}
\]

2) The 12th term of the sequence is:

\[
a_{12} = -708588
\]