find the indicated term of the geometric sequence a2=1/64, r=-2, n=7

To find the 7th term (\(a_7\)) of the geometric sequence, we will again use the formula for the nth term of a geometric sequence:

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

Given:
- \( a_2 = \frac{1}{64} \)
- \( r = -2 \)
- \( n = 7 \)

First, we need to find the first term (\(a_1\)) using the information provided for \(a_2\):

\[
a_2 = a_1 \cdot r
\]

Substitute the known values:

\[
\frac{1}{64} = a_1 \cdot (-2)
\]

To find \( a_1 \), solve for \( a_1 \):

\[
a_1 = \frac{1}{64} \div (-2) = -\frac{1}{128}
\]

Now that we have \( a_1 = -\frac{1}{128} \), we can find the 7th term \( a_7 \) using the formula:

\[
a_7 = a_1 \cdot r^{6}
\]

Substituting the values:

\[
a_7 = -\frac{1}{128} \cdot (-2)^{6}
\]

Since \( (-2)^6 = 64 \), we get:

\[
a_7 = -\frac{1}{128} \cdot 64 = -\frac{64}{128} = -\frac{1}{2}
\]

So, the 7th term \( a_7 \) is \(-\frac{1}{2}\).

Would you like any further explanation or details on this?

Here are five related questions you might find interesting:
1. How would you determine the first term if only a later term and the common ratio are given?
2. How do you calculate the sum of the first n terms in a geometric series?
3. What happens to the terms of a geometric sequence when the common ratio is negative?
4. How does the sign of the common ratio affect the sequence's behavior?
5. Can a geometric sequence have all terms positive if the common ratio is negative?

**Tip:** In a geometric sequence, if the common ratio is negative, the signs of the terms will alternate between positive and negative.

Learn how to find the 7th term (a7) of a geometric sequence with a2=1/64 and r=-2. This problem covers geometric sequences, including the formula for the nth term and calculation steps. Suitable for students in Grades 10-12.

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