Find the next number in the given number series: 1, 3, 10, 41, 206, ?

Introduction / Context:
This question involves a non linear number series where each new term is generated from the previous term using both multiplication and addition. These higher order patterns are common in competitive exams and test whether candidates can observe relations that depend on the position of each term in the sequence. Our task is to extend the series 1, 3, 10, 41, 206 by one more term.

Given Data / Assumptions:
- Given series: 1, 3, 10, 41, 206, ?- The sequence is strictly increasing, with gaps growing rapidly.- Each term is expected to be generated from the previous one using a systematic rule that depends on the term position.

Concept / Approach:
In rapidly growing series, a good first step is to check products and then see if a small constant is added or subtracted. When the multiplication factor does not remain fixed, it often depends on the position of the term. So we try a rule of the form a(n+1) = a(n) * k(n) + c, where k(n) depends on n. By experimenting with small integer values of k(n) linked to the term index, we can uncover the correct pattern.

Step-by-Step Solution:
- Let the first term be a1 = 1.- From 1 to 3: 1 * 2 + 1 = 3. Here the multiplier is 2.- From 3 to 10: 3 * 3 + 1 = 10. Here the multiplier is 3.- From 10 to 41: 10 * 4 + 1 = 41. Here the multiplier is 4.- From 41 to 206: 41 * 5 + 1 = 206. Here the multiplier is 5.- We observe a clear pattern: a(n+1) = a(n) * (n + 1) + 1, starting with n = 1.- To find the next term a6, we use the same rule with multiplier 6: a6 = a5 * 6 + 1.- Calculation: a6 = 206 * 6 + 1 = 1236 + 1 = 1237.- Therefore, the missing number in the series is 1237.

Verification / Alternative check:
- Rebuild the series using the discovered rule: a1 = 1; a2 = 1 * 2 + 1 = 3; a3 = 3 * 3 + 1 = 10; a4 = 10 * 4 + 1 = 41; a5 = 41 * 5 + 1 = 206; a6 = 206 * 6 + 1 = 1237.- All steps are consistent with the rule a(n+1) = a(n) * (n + 1) + 1, with no mismatches.

Why Other Options Are Wrong:
- 897, 987, 1007 and 823 do not equal 206 * 6 + 1, and none fits a simple positional multiplier pattern.- Using any of these values would destroy the elegant link between term index and multiplication factor.

Common Pitfalls:
- A common mistake is to look only at differences (2, 7, 31, 165) and try to detect a pattern there, which is much less obvious.- Some candidates also assume a constant multiplication factor, which cannot explain the transition from 1 to 3 to 10 to 41 to 206.- Ignoring the idea that the multiplier might depend on the position n can prevent recognition of the true rule.

Final Answer:
The series follows the rule a(n+1) = a(n) * (n + 1) + 1, so the next term is 206 * 6 + 1 = 1237.