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!