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.