Consider the following sequence of numbers: 1, 2, 3, 10, ?, 9802 Which number should replace the question mark so that the series follows a consistent pattern?

Introduction / Context:
This number series problem involves a non-trivial pattern, where each term is generated from the previous term using squaring combined with small adjustments. The challenge is to discover this pattern and then use it to find the missing value between 10 and 9802.

Given Data / Assumptions:

• Series: 1, 2, 3, 10, ?, 9802.
• Exactly one term (the fifth) is missing.
• The sequence is assumed to follow a consistent mathematical rule.
• Options for the missing term: 99, 199, 299, 999.

Concept / Approach:
Observing the series, squaring seems promising because the last term 9802 is very large compared to 99 or 199. We look for a pattern of the form next term = (current term)² ± 1. When we check this systematically, we see that the series alternates between adding 1 and subtracting 1 after squaring the previous term.

Step-by-Step Solution:
Step 1: From 1 to 2: 1² + 1 = 2. So the rule here is square then add 1. Step 2: From 2 to 3: 2² - 1 = 4 - 1 = 3. This time the rule is square then subtract 1. Step 3: From 3 to 10: 3² + 1 = 9 + 1 = 10. Again, it is square then add 1. Step 4: To continue the alternating pattern, the next step should be square then subtract 1. Step 5: So the missing term should be 10² - 1 = 100 - 1 = 99. Step 6: Confirm this pattern with the last term: from 99 to 9802, we should again square and add 1. Step 7: 99² + 1 = 9801 + 1 = 9802, which matches the final term given in the series.

Verification / Alternative check:
This alternating pattern (square and add 1, then square and subtract 1) fits every known transition in the series: 1 → 2, 2 → 3, 3 → 10 and 99 → 9802. Therefore 99 is uniquely determined as the missing term, and the rule holds consistently across the entire sequence.

Why Other Options Are Wrong:
If we choose 199, 299 or 999, the step from 10 to that value cannot be explained by a simple square ± 1 operation, nor can we then get exactly 9802 from that value using the same rule. These alternatives break the clean alternating pattern, so they are not acceptable as the missing term.

Common Pitfalls:
A frequent mistake is to look only for constant differences or fixed ratios, which do not appear here. Some learners also fail to notice that very large final terms often indicate squaring or similar exponential operations. Thinking in terms of square-and-adjust patterns is a useful habit for such series problems.

Final Answer:
The number that completes the pattern correctly is 99, so option (a) is the correct answer.