Study the following pattern :
9 × 9 = 81
99 × 99 = 9801
999 × 999 = 998001
.................................
What id 99999 × 99999 ?

Question

Study the following pattern :
9 × 9 = 81
99 × 99 = 9801
999 × 999 = 998001
.................................
What id  99999 × 99999 ?

Accepted Answer

To understand why Option 2 (9999800001) is the correct answer for the pattern 99999 × 99999, let's analyze the pattern and apply logical reasoning.

### Explanation of the Pattern:

1. **Pattern Observation:**
   - The given pattern is:
     - 9 × 9 = 81
     - 99 × 99 = 9801
     - 999 × 999 = 998001
   - We observe that as the number of 9s increases, the result follows a specific pattern.

2. **Pattern Analysis:**
   - For 9 × 9 = 81, the result is 81, which is 1 less than 100.
   - For 99 × 99 = 9801, the result is 9801, which is 99 less than 10000.
   - For 999 × 999 = 998001, the result is 998001, which is 999 less than 1000000.

3. **Generalizing the Pattern:**
   - The pattern can be generalized as follows: For a number consisting of 'n' 9s, the result of multiplying it by itself is a number that starts with 'n' 9s, followed by 'n-1' 8s, and ends with a 1.
   - This can be mathematically represented as: (10^n - 1) × (10^n - 1) = (10^(2n) - 2×10^n + 1).

4. **Applying the Pattern to 99999 × 99999:**
   - Here, n = 5 (since there are five 9s).
   - According to the pattern, the result should start with five 9s, followed by four 8s, and end with a 1.
   - Therefore, the result is 9999800001.

### Explanation of Why Other Options are Incorrect:

1. **Option 1: 9999888001**
   - This option suggests that the result has two 8s, which does not follow the pattern of having 'n-1' 8s. For five 9s, there should be four 8s, not two.

2. **Option 3: 999880001**
   - This option suggests that the result has only three 8s, which again does not follow the pattern. For five 9s, there should be four 8s.

3. **Option 4: 9999880001**
   - This option suggests that the result has three 8s, which is incorrect according to the pattern. The correct result should have four 8s.

In conclusion, Option 2 (9999800001) correctly follows the established pattern for multiplying numbers consisting entirely of 9s, making it the correct answer.