The given sequence is: \( 5, 10, \_\_, 50, 122, 170 \).

To determine the missing number, let's analyze the pattern in the sequence:

1. **First difference** between the consecutive terms:
   \[
   10 - 5 = 5
   \]
   \[
   \text{(missing term)} - 10
   \]
   \[
   50 - \text{(missing term)}
   \]
   \[
   122 - 50 = 72
   \]
   \[
   170 - 122 = 48
   \]

2. **Second difference** to find the pattern:
   - The difference between \(72\) and \(48\) is \(-24\).

Assuming the second difference is constant, the missing number should result in a sequence that fits this pattern. Let's hypothesize and check:
  
If we assume the first difference increases by the same amount between terms (in steps), then:

- Difference between the second and first differences:
   \[
   \text{72} - \text{first difference} = 72 - 24 = 48
   \]
   \[
   50 - \text{missing number} = -22
   \]
   If \(50 - 28\) is the next term, the number is closer to \(28\). This would mean the missing number should be \(28\) based on maintaining the \( -24 \) second difference.

Thus, the missing number is **28**.

---

Would you like more details or have any questions?

Here are five related questions for further exploration:

1. What is the next number in the sequence after 170?
2. Can you derive a formula for the \( n \)-th term of this sequence?
3. How would the pattern change if the second difference were a different constant?
4. What are the differences between other common sequences (like arithmetic vs. geometric)?
5. How does the method of finite differences help in identifying patterns?

**Tip:** In sequences, observing the first and second differences often reveals linear or quadratic patterns, helping to predict missing terms.

Explore how to find the missing number in the sequence 5, 10, ___, 50, 122, 170 using finite differences. This detailed solution includes step-by-step analysis of first and second differences, making it suitable for students in Grades 8-10.

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