If `1^3 = 1, 2^3 = 3 + 5, 3^3 = 7 + 9 + 11, 4^3 = 13 + 15 + 17 + 19, 5^3 = 21 + 23 + 25 + …. + 29`, etc. Then the value of `(100)^3` is equal to :

Options

A. 9901 + 9903 + …… + 10099
B. 9999 + 10001 + 10003+ ……+ 10199
C. 9989 + 9991 + …….+ 10089
D. Any two of (a) , (b) , (c)

Answer

To solve the problem, we need to find the value of \( (100)^3 \) based on the pattern given in the question. The pattern shows that each cube of a number \( n \) can be expressed as the sum of \( n \) consecutive odd numbers.

### Step-by-step Solution:

1. **Identify the Pattern**:
- From the given information:
- \( 1^3 = 1 \)
- \( 2^3 = 3 + 5 \)
- \( 3^3 = 7 + 9 + 11 \)
- \( 4^3 = 13 + 15 + 17 + 19 \)
- \( 5^3 = 21 + 23 + 25 + 27 + 29 \)
- We can see that the first term of each series of odd numbers increases as \( n \) increases.

2. **Determine the First Odd Number for \( n \)**:
- The first term of the series for \( n \) can be derived from the formula:
\[ \text{First term} = n^2 - n + 1 \]
- Let's verify this for the first few values of \( n \):
- For \( n = 1 \): \( 1^2 - 1 + 1 = 1 \)
- For \( n = 2 \): \( 2^2 - 2 + 1 = 3 \)
- For \( n = 3 \): \( 3^2 - 3 + 1 = 7 \)
- For \( n = 4 \): \( 4^2 - 4 + 1 = 13 \)
- For \( n = 5 \): \( 5^2 - 5 + 1 = 21 \)

3. **Calculate the First Term for \( n = 100 \)**:
- Using the formula for \( n = 100 \):
\[ \text{First term} = 100^2 - 100 + 1 = 10000 - 100 + 1 = 9901 \]

4. **Sum of Odd Numbers**:
- The sum of the first \( n \) odd numbers starting from \( 9901 \) can be expressed as:
\[ \text{Sum} = 9901 + (9901 + 2) + (9901 + 4) + ... + (9901 + 2 \times (100 - 1)) \]
- This is the sum of \( 100 \) odd numbers starting from \( 9901 \).

5. **Calculate the Last Term**:
- The last term in this series would be:
\[ \text{Last term} = 9901 + 2 \times (100 - 1) = 9901 + 198 = 10099 \]

6. **Calculate the Total Sum**:
- The total sum of an arithmetic series can be calculated using the formula:
\[ \text{Sum} = \frac{\text{Number of terms}}{2} \times (\text{First term} + \text{Last term}) \]
- Here, the number of terms is \( 100 \):
\[ \text{Sum} = \frac{100}{2} \times (9901 + 10099) = 50 \times 20000 = 1000000 \]

### Final Result:
Thus, the value of \( (100)^3 \) is \( 1000000 \).