12, 6, 3, 3/2, 3/4  find the sum of the infinite geometric series

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 \( |r| < 1 \).

12, 6, 3, 3/2, 3/4 find the sum of the infinite geometric series

Learn how to find the sum of the infinite geometric series 12, 6, 3, 3/2, 3/4 using the formula S_∞ = a / (1 - r). This detailed solution covers algebraic concepts and shows how the series converges when the common ratio is less than 1. Suitable for students in Grades 9-12.

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