Find the missing number in the following series: 2, 3, 10, 39, ?, 885.

Introduction / Context:

This is a number series problem where the terms grow rapidly, suggesting a combination of multiplication and addition. Such questions are intended to test a deeper understanding of series in which each term is generated from the previous one using both multiplication and an additional term that itself follows a pattern. Recognizing this two level structure is essential for many aptitude exams involving complex numerical reasoning.

Given Data / Assumptions:

• The series is 2, 3, 10, 39, ?, 885.
• We need to find a single missing term between 39 and 885.
• The numbers increase quickly, hinting at repeated multiplication by a variable factor.
• There is likely a secondary pattern in the additive component, such as squares or cubes of integers.

Concept / Approach:

When growth is fast but not purely exponential, a common pattern is a mix of multiplication and addition. Specifically, many exam series follow a rule like a(n+1) = a(n) * k(n) + extra(n), where k(n) and extra(n) follow simple sequences. A good starting point is to try multiplying each term by small integers and then checking the gap between the product and the next term. If those gaps form a recognizable pattern, such as consecutive squares, we can generalize the rule and compute the missing term accordingly.

Step-by-Step Solution:

Step 1: From 2 to 3: 2 * 1 = 2 and 2 + 1 = 3, so effectively 2 * 1 + 1 = 3. Step 2: From 3 to 10: 3 * 2 = 6 and 6 + 4 = 10, so 3 * 2 + 4 = 10. Step 3: From 10 to 39: 10 * 3 = 30 and 30 + 9 = 39, so 10 * 3 + 9 = 39. Step 4: Notice that the added numbers 1, 4, 9 are 1^2, 2^2, 3^2 respectively. Step 5: Continue the pattern: next step should be 39 * 4 + 4^2 = 39 * 4 + 16 = 156 + 16 = 172, and then 172 * 5 + 5^2 = 172 * 5 + 25 = 860 + 25 = 885, which matches the last term.

Verification / Alternative check:

Check each transition with the rule a(n+1) = a(n) * n + n^2 starting from n = 1. For n = 1: 2 * 1 + 1^2 = 3. For n = 2: 3 * 2 + 2^2 = 10. For n = 3: 10 * 3 + 3^2 = 39. For n = 4: 39 * 4 + 4^2 = 172. For n = 5: 172 * 5 + 5^2 = 885. Every step fits perfectly, including the given final term 885, which confirms that 172 is the only consistent missing value.

Why Other Options Are Wrong:

Option 128, when used as the missing term, fails because 128 * 5 + 25 equals 665, not 885, breaking the sequence. Option 156 leads to 156 * 5 + 25 = 805, which does not match. Option 189 produces 189 * 5 + 25 = 970, again inconsistent. Option 164 gives 164 * 5 + 25 = 845, still not equal to 885. None of these maintain the rule of multiplication by increasing integers plus perfect squares, so they must be rejected.

Common Pitfalls:

Many learners initially look only at differences between terms, which in this case are 1, 7, 29, and then some unknown gap, and do not see a simple pattern. Others might attempt fixed multiplication factors without considering an additive part. A further trap is to guess the missing term using approximate growth rather than exact rules. The safe strategy is to check for multiplying by 1, 2, 3, 4 and adding 1^2, 2^2, 3^2, 4^2, which is a familiar exam pattern that often appears in high level reasoning questions.

Final Answer:

The missing number that follows the rule a(n+1) = a(n) * n + n^2 and leads correctly to 885 is 172.