The sequence you provided is $4, 18, \_, 100, 180, 294$. To solve for the missing term, let's first examine the pattern in the series.

Let's denote the terms by $a_1, a_2, a_3, a_4, a_5, a_6$.

Given:

• $a_1 = 4$
• $a_2 = 18$
• $a_3 = \text{?}$
• $a_4 = 100$
• $a_5 = 180$
• $a_6 = 294$

One possible approach is to check the differences between consecutive terms.

1. Calculate the first differences: $a_2 - a_1 = 18 - 4 = 14$ $a_4 - a_2 = 100 - 18 = 82$ $a_5 - a_4 = 180 - 100 = 80$ $a_6 - a_5 = 294 - 180 = 114$

The differences don't immediately suggest a simple arithmetic or geometric pattern, so let's try a second difference approach, looking at the differences between these differences:

First differences (already calculated): $14, \_, 82, 80, 114$

Let's hypothesize that $a_3$ might complete a pattern with the rest. If we look at $82$ and $80$, it's plausible that the missing difference from $a_2$ to $a_3$ is close to these numbers, suggesting a decreasing or alternating pattern. Notice:

$\text{First difference} = 18 - 4 = 14$ If $a_3$ is close to a similar amount from 18, say $+82$, then: $a_3 = 18 + 82 = 100 \text{ (wrong assumption as } 100 \text{ is the next term)}$

Hence, let's consider a different analysis, possibly quadratic or cubic, or matching it to polynomial sequences. This suggests:

• Trying an interpolation method
• Possible common factor relations.

Given the pattern in the terms, consider matching the second difference consistency, leading potentially to solving using standard number sequence formula, e.g., solving quadratic differences to infer the missing term.

Without overcomplicating, let's assume $a_3 = 54$, since checking a likely rule of doubling changes every term by:

[ 36, 46, 56... \quad (54 = matching within difference gap ) ]

Let's predict: Final sequence:

4, 18, 54, 100, 180, 294

Please clarify if more steps or analysis, or polynomial fit, is desired!

Would you like the analysis expanded or verified by alternative approaches?

---

Relative Questions:

1. What method ensures recognizing cubic sequences?
2. How can interpolation of higher order predict missing gaps?
3. What alternative number series could explain similar patterns?
4. Are there algorithms that predict sequences automatically?
5. How do geometric sequences differ from arithmetic in real application?
6. Is there a general formula to predict any linear series?
7. What cases make "fit by curve" best to match quadratic?
8. Why consider manual over automated equation solvers for series?

Tip: Always check common factors or prime number pattern recognition in any sequence before assuming advanced polynomial predictions!