In the number series 7, 8, 18, 57, ?, 1165, 6996, one term is missing. Find the number that should replace the question mark to maintain the pattern.

Introduction / Context:
This problem features a rapidly increasing number series: 7, 8, 18, 57, ?, 1165, 6996. Such explosive growth almost always indicates multiplicative patterns, often of the form previous term times a factor plus or minus something. Our task is to discover this rule and fill in the missing number.

Given Data / Assumptions:

• Series: 7, 8, 18, 57, ?, 1165, 6996.
• Exactly one term is missing between 57 and 1165.
• The pattern is expected to be consistent across the entire series.

Concept / Approach:
We look for a relation of the type a_n+1 = a_n * k + k, where k changes in a simple way like 1, 2, 3, 4, etc. This is a common design in competitive exams: multiply by an increasing integer and then add the same integer. We test this hypothesis on the known consecutive terms.

Step-by-Step Solution:
1. From 7 to 8: 7 * 1 + 1 = 7 + 1 = 8. 2. From 8 to 18: 8 * 2 + 2 = 16 + 2 = 18. 3. From 18 to 57: 18 * 3 + 3 = 54 + 3 = 57. 4. We see a clear pattern: multiply by 1, 2, 3, and each time add the same multiplier. 5. Next step should therefore multiply by 4 and add 4: 57 * 4 + 4 = 228 + 4 = 232. 6. So the missing term is 232. 7. Confirm the same logic for the next term: 232 * 5 + 5 = 1160 + 5 = 1165. 8. Next again: 1165 * 6 + 6 = 6990 + 6 = 6996. 9. The pattern holds throughout, so 232 is correct.

Verification / Alternative check:
Rebuild the full series using the discovered rule: Start: 7. Step 1: 7 * 1 + 1 = 8. Step 2: 8 * 2 + 2 = 18. Step 3: 18 * 3 + 3 = 57. Step 4: 57 * 4 + 4 = 232. Step 5: 232 * 5 + 5 = 1165. Step 6: 1165 * 6 + 6 = 6996. Each transition follows the same multiply by n, then add n rule with n increasing from 1 to 6.

Why Other Options Are Wrong:

• 228, 542, 415: None of these values maintain the structure a_n+1 = a_n * n + n with n taking integer values 1, 2, 3, 4, 5, 6.
• Substituting any of those numbers causes the jump to 1165 to fail the pattern, so they are inconsistent.

Common Pitfalls:
Without trying multiplicative patterns, students may attempt simple differences, which appear chaotic and give no simple rule. Another common error is to try constant multiplication factors instead of allowing the factor to increase like 1, 2, 3, and so on. Recognising that both multiplication and addition by the same number are involved is the key insight here.

Final Answer:
The number that correctly completes the series is 232.