https://file.aiquickdraw.com/mathbot/file/1726757933738o8f8mhx4.jpg

The chalkboard in the image contains four different sequences, and the task seems to involve determining whether these infinite sequences converge to a sum or not. Let’s break them down:

1. **First Sequence:**  
   \( 4 + 1 + \frac{1}{4} + \dots \)  
   This is likely a geometric series where the first term is 4, and the common ratio is decreasing. We would need to determine the common ratio for the series to check if the sum converges or diverges.

2. **Second Sequence:**  
   \( 16 + 8 + 4 + 2 + \dots \)  
   This is another geometric series with the first term \(a = 16\) and the common ratio \(r = \frac{1}{2}\). This sequence converges because the common ratio is less than 1. The sum of an infinite geometric series is given by:  
   \[
   S = \frac{a}{1 - r} = \frac{16}{1 - \frac{1}{2}} = 32
   \]

3. **Third Sequence:**  
   \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots \)  
   This is a geometric series where \(a = 1\) and \(r = \frac{1}{2}\). Since the common ratio is less than 1, it converges. The sum of this geometric series is:
   \[
   S = \frac{a}{1 - r} = \frac{1}{1 - \frac{1}{2}} = 2
   \]

4. **Fourth Sequence:**  
   \( 4 + 2 + 1 + \dots \)  
   Another geometric series with \(a = 4\) and \(r = \frac{1}{2}\). Like the second and third sequences, this one converges as well, and its sum is:  
   \[
   S = \frac{a}{1 - r} = \frac{4}{1 - \frac{1}{2}} = 8
   \]

It seems the third sequence was labeled incorrectly with "No sum" (but it does converge). 

Would you like further explanation on any of these sequences?

Here are 5 related questions for deeper understanding:
1. How do you derive the formula for the sum of an infinite geometric series?
2. Why does the geometric series converge only when the common ratio is less than 1?
3. Can you give an example of a divergent series and explain why it doesn’t converge?
4. How do the results change if the first term of a geometric series is negative?
5. What are the applications of geometric series in real-world contexts?

**Tip:** Always check the common ratio of a geometric series to determine if it converges or diverges—convergence only happens if the absolute value of the ratio is less than 1.

Determine if the infinite sequences on the chalkboard have a sum.

Explore how to determine whether the given infinite sequences have a sum. This problem involves geometric series with different terms and common ratios, suitable for Grades 9-12. Includes step-by-step explanations and key formulas.

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