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The sequence given is \( 272, -136, 68, -34, \dots \), which appears to be a geometric series.

To confirm, let's determine if there is a common ratio, \( r \), between consecutive terms:

\[
r = \frac{-136}{272} = -\frac{1}{2}
\]

\[
r = \frac{68}{-136} = -\frac{1}{2}
\]

\[
r = \frac{-34}{68} = -\frac{1}{2}
\]

Since the ratio is consistently \( r = -\frac{1}{2} \), this is indeed a geometric sequence.

### Formula for the Sum of an Infinite Geometric Series:
The sum \( S \) of an infinite geometric series can be found using the formula:

\[
S = \frac{a}{1 - r}
\]

where:
- \( a = 272 \) (the first term),
- \( r = -\frac{1}{2} \) (the common ratio).

Substituting these values into the formula:

\[
S = \frac{272}{1 - \left(-\frac{1}{2}\right)} = \frac{272}{1 + \frac{1}{2}} = \frac{272}{\frac{3}{2}} = 272 \times \frac{2}{3} = \frac{544}{3}
\]

Thus, the sum of the infinite geometric series is:

\[
S = \frac{544}{3}
\]

Would you like more details or have any further questions?

Here are 5 related questions to explore:

1. How do you determine if a sequence is geometric?
2. What are the conditions for an infinite geometric series to converge?
3. Can an infinite series have a finite sum? How is that possible?
4. What happens if the common ratio is greater than 1 in magnitude?
5. How do you sum a finite geometric series?

**Tip**: Always check if the absolute value of the common ratio is less than 1 when summing an infinite geometric series!

Find the sum of the sequence 272, -136, 68, -34, ......... Express your answer as a fraction in its simplest form.

Find the sum of the infinite geometric sequence 272, -136, 68, -34,... using the sum formula for geometric series. Learn key concepts about sequences, series, and their sums. Suitable for students in Grades 9-12.

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