Find the missing number in the series: 2, 3, 8, ?, 112, 565

Introduction / Context:
This number series grows very quickly and includes both multiplication and addition at each step. Such patterns often depend on the position of the term, with each step using the term index as a multiplier or an additive constant. The given sequence is 2, 3, 8, ?, 112, 565, and we must determine the missing value that makes the pattern consistent.

Given Data / Assumptions:
- Known terms: 2, 3, 8, ?, 112, 565.- One intermediate term between 8 and 112 is missing.- The series is strictly increasing with very rapid growth.- The rule likely involves multiplying by the step number and then adding that same step number.

Concept / Approach:
When numbers increase very fast, we check whether each term can be written as previous term * k + c. If k and c change in a simple way linked to the step index, we can reconstruct the rule. Here, we will test a pattern of the form a(n+1) = a(n) * n + n, where n increases as we move through the series.

Step-by-Step Solution:
- Label the terms: a1 = 2, a2 = 3, a3 = 8, a4 = ?, a5 = 112, a6 = 565.- From a1 to a2: 2 * 1 + 1 = 2 + 1 = 3.- From a2 to a3: 3 * 2 + 2 = 6 + 2 = 8.- This suggests the rule a(n+1) = a(n) * n + n.- For a4, use n = 3: a4 = a3 * 3 + 3 = 8 * 3 + 3 = 24 + 3 = 27.- For a5, use n = 4: a5 = a4 * 4 + 4 = 27 * 4 + 4 = 108 + 4 = 112.- For a6, use n = 5: a6 = a5 * 5 + 5 = 112 * 5 + 5 = 560 + 5 = 565.- All transitions match perfectly, confirming that the missing term is 27.

Verification / Alternative check:
- Reconstructed series: 2, 3, 8, 27, 112, 565.- At each step, we multiply by the current index and add the same index: times 1 plus 1, times 2 plus 2, times 3 plus 3, times 4 plus 4, times 5 plus 5.- No contradictions appear, so the rule is fully consistent.

Why Other Options Are Wrong:
- 565 and 112 are later terms in the series, not the missing intermediate value between 8 and 112.- 8 and 16 cannot satisfy the positional rule a(n+1) = a(n) * n + n when placed at the fourth position.- Using any value other than 27 breaks the multiplication and addition pattern tied to the term index.

Common Pitfalls:
- Some candidates look only at differences, which appear irregular here and do not reveal the pattern.- Ignoring the possibility that the step number itself enters the rule leads to incorrect attempts with constant multipliers.- Confusing the roles of multiplication and addition in the rule can cause calculation mistakes.

Final Answer:
The series follows a(n+1) = a(n) * n + n, so the missing term is 27.