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The image you've uploaded provides instructions on how to find the nth term in a geometric sequence. Here’s a detailed breakdown:

### Geometric Sequences: Finding the nth Term

#### Formula
The formula for finding the nth term in a geometric sequence is:

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

- \(a_n\): The nth term of the sequence.
- \(a_1\): The first term of the sequence.
- \(r\): The common ratio.
- \(n\): The term number.

#### Steps for Finding the nth Term:

1. **Find the Common Ratio (r)**:
   - Divide any term in the geometric sequence by the term preceding it:
     \[
     r = \frac{a_{n}}{a_{n-1}}
     \]

2. **Substitute Values into the Formula**:
   - Use the values you’ve found to calculate the desired term using the formula \(a_n = a_1 \times r^{(n-1)}\).

#### Examples:

- **Example 1:**
  - First term \(a_1\) = 2
  - Common ratio \(r\) = 3
  - Find the 5th term:
    \[
    a_5 = 2 \times 3^{(5-1)} = 2 \times 3^4 = 2 \times 81 = 162
    \]

- **Example 2:**
  - First term \(a_1\) = 64
  - Common ratio \(r\) = \frac{1}{2}
  - Find the 7th term:
    \[
    a_7 = 64 \times \left(\frac{1}{2}\right)^{(7-1)} = 64 \times \left(\frac{1}{2}\right)^6 = 64 \times \frac{1}{64} = 1
    \]

### Additional Questions to Explore:
1. How do you determine the common ratio if you're given two non-consecutive terms in a geometric sequence?
2. What happens to the nth term if the common ratio \(r\) is less than 1?
3. Can a geometric sequence have a common ratio of 0? What would be the implications?
4. How do you find the sum of the first n terms of a geometric sequence?
5. How does the geometric sequence formula change if the sequence is decreasing?

**Tip:** Always ensure that the common ratio is consistent throughout the sequence by checking multiple terms. This will help avoid errors when applying the formula.

How do I find the nth term in a geometric sequence?

Learn how to find the nth term in a geometric sequence using the formula a_n = a_1 × r^(n-1). This method helps calculate any term in the sequence given the first term and the common ratio. Suitable for students in Grades 8-10.

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