Question

the first term of a geometric sequence is 6 and the common ratio is -8.
1) create an equation for the situation
2) find the 7th term of the sequence

Answer 1

1) In a geometric sequence, the \( n \)-th term can be represented as:

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

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

For this situation, the first term \( a_1 = 6 \) and the common ratio \( r = -8 \). Therefore, the equation for the \( n \)-th term of the sequence is:

\[
a_n = 6 \cdot (-8)^{(n-1)}
\]

2) To find the 7th term of the sequence \( a_7 \):

\[
a_7 = 6 \cdot (-8)^{(7-1)} = 6 \cdot (-8)^6
\]

Now, we calculate \( (-8)^6 \):

\[
(-8)^6 = 262144
\]

So,

\[
a_7 = 6 \cdot 262144 = 1572864
\]

Thus, the 7th term of the sequence is:

\[
\boxed{1572864}
\]