Question

the first term of a geometric sequence is -3 and the common ratio is 1/2.
1) create an equation for the situation a_n=
2) find the 6th term of the sequence, give answer as a fraction.

Answer 1

In a geometric sequence, the general formula for the \(n\)-th term \(a_n\) is given by:

\[
a_n = a_1 \cdot r^{(n-1)}
\]

where:
- \(a_1\) is the first term,
- \(r\) is the common ratio,
- \(n\) is the term number.

Given that the first term \(a_1 = -3\) and the common ratio \(r = \frac{1}{2}\), we can write the equation for the \(n\)-th term as follows:

1) The equation for the geometric sequence is:

\[
a_n = -3 \cdot \left(\frac{1}{2}\right)^{(n-1)}
\]

Now, to find the 6th term of the sequence, we substitute \(n = 6\) into the equation:

2) Calculate \(a_6\):

\[
a_6 = -3 \cdot \left(\frac{1}{2}\right)^{(6-1)} = -3 \cdot \left(\frac{1}{2}\right)^{5}
\]

Now, calculate \(\left(\frac{1}{2}\right)^{5}\):

\[
\left(\frac{1}{2}\right)^{5} = \frac{1}{2^5} = \frac{1}{32}
\]

Now substitute back into the equation:

\[
a_6 = -3 \cdot \frac{1}{32} = -\frac{3}{32}
\]

Therefore, the 6th term of the sequence is:

\[
\boxed{-\frac{3}{32}}
\]