To find the sum of an infinite geometric series, we use the formula:

$S_{\infty} = \frac{a}{1 - r}$

where:

• $a$ is the first term of the series,
• $r$ is the common ratio, and
• $|r| < 1$ (this ensures the series converges).

Step 1: Identify the first term and common ratio.

• The first term, $a = 12$.
• To find the common ratio $r$, divide the second term by the first term: $r = \frac{6}{12} = \frac{1}{2}$

Step 2: Apply the formula for the sum of an infinite geometric series.

Since $r = \frac{1}{2}$, the series converges, and we can calculate the sum:

$S_{\infty} = \frac{12}{1 - \frac{1}{2}} = \frac{12}{\frac{1}{2}} = 12 \times 2 = 24$

Final Answer:

The sum of the infinite geometric series is $24$.

Would you like further details or have any questions?

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Here are 5 related questions to expand on this topic:

1. What happens if the common ratio $r$ is greater than or equal to 1?
2. How do we find the sum of a finite geometric series?
3. Can an infinite series have a negative common ratio? If so, how does it affect the sum?
4. What is the common ratio if the second term of the geometric series was 9 instead of 6?
5. How does the geometric series formula change for series starting with a different term than 1?

Tip: Always verify the common ratio before applying the infinite series formula, as the series converges only when $|r| < 1$.