Find the sum to infinity of the geometric series 32 + 16 + 8 + ….

To find the sum to infinity of a geometric series, we use the formula:

\[
S_\infty = \frac{a}{1 - r}
\]

where:
- \( a \) is the first term,
- \( r \) is the common ratio, with \( |r| < 1 \) for the sum to converge.

In this series:
- The first term \( a = 32 \),
- The common ratio \( r = \frac{16}{32} = \frac{1}{2} \).

Now we can substitute these values into the formula:

\[
S_\infty = \frac{32}{1 - \frac{1}{2}} = \frac{32}{\frac{1}{2}} = 32 \times 2 = 64
\]

### Answer:
The sum to infinity of the series is \( 64 \).

Would you like a detailed breakdown of these steps or have any questions?

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1. How do we derive the formula for the sum of an infinite geometric series?
2. Why must the common ratio \( r \) be less than 1 in magnitude for convergence?
3. How would we calculate the sum if the series was finite?
4. Can a geometric series with a common ratio greater than 1 have a sum to infinity?
5. What happens to the sum if the common ratio \( r \) is negative?

**Tip:** In a geometric series, if the common ratio is a fraction less than 1, the series will converge, making the infinite sum possible.

Discover how to find the sum to infinity of the geometric series 32 + 16 + 8 + …. This solution provides a detailed step-by-step guide to using the sum to infinity formula S_∞ = a / (1 - r), explaining key concepts like series convergence. Suitable for Grades 10-12.

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