find the eighth term of the geometric sequence 120 60 30 15

To find the eighth term of a geometric sequence, we need to identify the first term (\(a\)) and the common ratio (\(r\)) of the sequence.

### Step 1: Identify the First Term and Common Ratio
- The first term (\(a\)) is given as \(120\).
- To find the common ratio (\(r\)), we can divide the second term by the first term:

\[
r = \frac{\text{second term}}{\text{first term}} = \frac{60}{120} = \frac{1}{2}
\]

### Step 2: Use the Formula for the \(n\)-th Term
The formula for the \(n\)-th term (\(a_n\)) of a geometric sequence is:

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

To find the eighth term (\(a_8\)), we set \(n = 8\):

\[
a_8 = 120 \cdot \left(\frac{1}{2}\right)^{7}
\]

### Step 3: Calculate the Eighth Term
\[
a_8 = 120 \cdot \left(\frac{1}{2^7}\right) = 120 \cdot \frac{1}{128}
\]

\[
a_8 = \frac{120}{128}
\]

\[
a_8 = \frac{15}{16}
\]

Therefore, the eighth term of the geometric sequence is \(\frac{15}{16}\).

Would you like more details or have any questions?

### Related Questions:

1. How would you find the tenth term of the same sequence?
2. What is the sum of the first eight terms of this sequence?
3. How do you determine the common ratio in a geometric sequence?
4. What happens if the common ratio is negative? How does it affect the sequence?
5. How would you find the general term (\(a_n\)) for this sequence?

**Tip:** In a geometric sequence, each term is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio.

Learn how to find the eighth term of the geometric sequence 120, 60, 30, 15 using the formula for the n-th term of a geometric sequence. This problem involves identifying the first term and common ratio of the sequence and applying the formula step-by-step. Suitable for students in Grades 7-9.

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