In the number series 10000, 11000, 9900, 10890, ?, 10781, one term is missing. Find the number that should replace the question mark to maintain the pattern.

Introduction / Context:
The given series 10000, 11000, 9900, 10890, ?, 10781 contains relatively large numbers and one missing term. The pattern suggests alternating increases and decreases of around ten percent, which is a common structure in reasoning questions. We aim to confirm this and identify the missing number.

Given Data / Assumptions:

• Series: 10000, 11000, 9900, 10890, ?, 10781.
• Exactly one term between 10890 and 10781 is missing.
• The first term is 10000 and the pattern appears to involve percentage changes.

Concept / Approach:
Notice that 11000 is 10 percent more than 10000, and 9900 is 10 percent less than 11000. This suggests an alternating pattern of increasing by 10 percent (multiply by 1.1) and decreasing by 10 percent (multiply by 0.9). We apply this logic step by step to find the missing term.

Step-by-Step Solution:
1. Start from 10000. 2. Increase by 10 percent to get the next term: 10000 * 1.1 = 11000. 3. Decrease 11000 by 10 percent to get the third term: 11000 * 0.9 = 9900. 4. Again increase 9900 by 10 percent: 9900 * 1.1 = 10890. 5. Now we must decrease 10890 by 10 percent to get the missing fifth term: 10890 * 0.9 = 9801. 6. Finally, increase 9801 by approximately 10 percent for the last given term: 9801 * 1.1 = 10781.1 which is essentially 10781 when rounded. 7. This confirms that the pattern alternates: +10 percent, -10 percent, +10 percent, -10 percent, and so on.

Verification / Alternative check:
Write out the entire series using this rule: Start with 10000. 10000 * 1.1 = 11000. 11000 * 0.9 = 9900. 9900 * 1.1 = 10890. 10890 * 0.9 = 9801. 9801 * 1.1 ≈ 10781. All terms either match exactly or within natural rounding, which is acceptable in aptitude problems of this type.

Why Other Options Are Wrong:

• 10423, 10241, 9712: None of these values equal exactly 90 percent of 10890. Substituting them breaks the consistent pattern of alternating 10 percent increase and 10 percent decrease.

Common Pitfalls:
A common error is to look only at absolute differences rather than percentage changes. The differences 1000, 1100, 990, etc., seem irregular unless we notice that they are proportional to the current term. Recognising that the operations are multiplicative (1.1 or 0.9) rather than additive is crucial here.

Final Answer:
The missing number in the series is 9801.