The equation you've provided represents the sum $S_n$ of the first $n$ terms of a geometric sequence. Let's break down the formula step by step:

The sum of the first $n$ terms of a geometric sequence is given by:

$S_n = a_1 + a_1 r + a_1 r^2 + \dots + a_1 r^{n-1}$

Where:

• $S_n$ is the sum of the first $n$ terms of the geometric sequence.
• $a_1$ is the first term of the geometric sequence.
• $r$ is the common ratio between consecutive terms.
• $n$ is the number of terms to be summed.

Deriving the Formula for $S_n$

This sum is actually a geometric series. To find a formula for this sum, we can use the standard formula for the sum of the first $n$ terms of a geometric series:

$S_n = \frac{a_1 (1 - r^n)}{1 - r} \quad \text{for} \quad r \neq 1$

Where:

• $a_1$ is the first term of the sequence.
• $r$ is the common ratio.
• $n$ is the number of terms.

This formula holds as long as $r \neq 1$. When $r = 1$, the sum simplifies to:

$S_n = a_1 \cdot n$

Explanation:

• For $r > 1$, the terms grow exponentially.
• For $0 < r < 1$, the terms shrink towards zero as $n$ increases.
• If $r = -1$, the sum alternates between positive and negative terms.
• If $r = 1$, all the terms are equal to $a_1$, so the sum is simply $a_1 \times n$.

Would you like me to explain any particular part of this or expand on examples?

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Here are 5 related questions to explore further:

1. How do you calculate the sum of a geometric series when $r = 1$?
2. What happens to the sum of a geometric sequence if the common ratio is negative?
3. How can you derive the formula for $S_n$ from the geometric series sum formula using induction?
4. How does the sum $S_n$ behave as $n \to \infty$ for different values of $r$?
5. What is the sum of an infinite geometric series, and how does it relate to the formula for $S_n$?

Tip: When dealing with geometric sequences, always remember that the common ratio $r$ plays a critical role in determining the sum's behavior. If $|r| > 1$, the series grows rapidly; if $|r| < 1$, the series converges towards a fixed value as $n$ increases.