A geometric progression (or geometric sequence) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the *common ratio*.

### Key Concepts in a Geometric Progression

1. **First Term (a):** This is the starting number of the sequence.
2. **Common Ratio (r):** This is the factor by which we multiply each term to get the next one.
   - If \( r > 1 \), the terms grow larger, creating a positive growth.
   - If \( 0 < r < 1 \), the terms get smaller, approaching zero.
   - If \( r < 0 \), the sequence alternates between positive and negative values.

3. **General Formula for the \( n \)-th Term:**
   The \( n \)-th term of a geometric sequence can be calculated as:
   \[
   a_n = a \times r^{(n-1)}
   \]
   where:
   - \( a \) is the first term,
   - \( r \) is the common ratio,
   - \( n \) is the term's position in the sequence.

4. **Sum of a Finite Geometric Sequence:**
   The sum of the first \( n \) terms of a geometric sequence (for \( r \neq 1 \)) is given by:
   \[
   S_n = a \frac{1 - r^n}{1 - r}
   \]

5. **Sum of an Infinite Geometric Sequence (when \( |r| < 1 \)):**
   For an infinite geometric sequence, the sum converges if \( |r| < 1 \) and is calculated as:
   \[
   S_{\infty} = \frac{a}{1 - r}
   \]

### Examples of Geometric Progressions

1. **Example 1:** For a sequence with \( a = 3 \) and \( r = 2 \):
   - The sequence is: \( 3, 6, 12, 24, 48, \dots \)
   - Here, each term is obtained by multiplying the previous term by \( 2 \).

2. **Example 2:** For a sequence with \( a = 5 \) and \( r = -0.5 \):
   - The sequence is: \( 5, -2.5, 1.25, -0.625, \dots \)
   - Here, each term is obtained by multiplying by \( -0.5 \), causing the terms to alternate in sign.

### Practical Applications

Geometric progressions are used in various fields, such as finance (compound interest), computer science (algorithm analysis), and physics (radioactive decay).

Would you like to see more examples or solve a specific problem?

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1. What is the formula for the sum of an infinite geometric sequence?
2. How does the common ratio affect the growth of a geometric sequence?
3. Can a geometric sequence have a common ratio of 1?
4. How would you derive the general formula for the \( n \)-th term in a geometric sequence?
5. How do geometric progressions differ from arithmetic progressions?

**Tip:** If a sequence is multiplying or dividing by a consistent number, it’s likely geometric!

Explore the fundamentals of geometric progression, including the n-th term formula, sum of finite and infinite geometric sequences, and practical applications. Ideal for students in Grades 9-12.

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