Can you find the ? in this series: 8, 16, ?, 110, -935

To find the missing number in the series \( 8, 16, ?, 110, -935 \), let's analyze the pattern.

### Step 1: Identify Differences
We will first calculate the differences between the consecutive numbers we know:
- \( 16 - 8 = 8 \)
- \( 110 - ? = ? \)  (unknown yet)
- \( -935 - 110 = -1045 \)

So the differences between the known numbers so far are: 
\[ +8, ?, -1045 \]

### Step 2: Look for a Potential Second-Difference Pattern
Often, sequences involve second differences (the difference of the differences). Let's compute them:
- The difference between 8 and the unknown difference \( ? \), and then check the consistency with the last difference.

### Step 3: Test Geometric or Recursive Patterns
If we assume the differences follow a certain geometric pattern (e.g., increasing or decreasing by a constant factor), we can attempt further analysis.

Let me compute the middle value.The first difference is \( +8 \), and the last difference is \( -1045 \). These differences suggest a sharp transition in the pattern, potentially exponential or quadratic.

Let's explore the possibility of a quadratic pattern for the sequence by solving for the missing term using this assumption.The solution to the quadratic pattern is:

\[
A = 13, \quad B = -31, \quad C = 26
\]

This gives the quadratic equation:

\[
a_n = 13n^2 - 31n + 26
\]

### Step 4: Find the Missing Term
Now, using this equation, we can calculate the third term (where \( n = 3 \)).

Let me compute it for you.The missing number in the series is \( 50 \). 

Thus, the complete series is:
\[
8, 16, 50, 110, -935
\]

Would you like further details on how the quadratic pattern was determined?

Here are five follow-up questions:
1. What happens if we try a different type of progression for the sequence?
2. How can we generalize this approach to other missing-term sequences?
3. Could this sequence have a deeper mathematical significance?
4. What are some common types of number patterns seen in mathematics?
5. How can higher-order differences help identify missing numbers?

**Tip:** Quadratic patterns arise often in sequences where the second differences between consecutive terms are constant.

Learn how to find the missing number in the sequence 8, 16, ?, 110, -935 using quadratic patterns and second differences. This problem covers key algebra concepts and solving quadratic equations, suitable for students in Grades 9-12.

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