Consider the following statements:
1. The sum of cubes of first 20 natural numbers is 44400.
2. The sum of squares of first 20 natural numbers is 2870.
Which of the above statement is/are correct?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Explanation

To determine the correctness of the given statements regarding the sums of cubes and squares of the first 20 natural numbers, we will calculate both sums step by step.

### Step 1: Calculate the sum of cubes of the first 20 natural numbers

The formula for the sum of cubes of the first \( n \) natural numbers is given by:

\[ \left( \frac{n(n + 1)}{2} \right)^2 \]

For \( n = 20 \):

\[ \text{Sum of cubes} = \left( \frac{20 \times (20 + 1)}{2} \right)^2 \]

Calculating the expression inside the parentheses:

\[ = \frac{20 \times 21}{2} = \frac{420}{2} = 210 \]

Now, squaring this result:

\[ 210^2 = 44100 \]

Thus, the sum of cubes of the first 20 natural numbers is \( 44100 \), not \( 44400 \). Therefore, the first statement is incorrect.

### Step 2: Calculate the sum of squares of the first 20 natural numbers

The formula for the sum of squares of the first \( n \) natural numbers is given by:

\[ \frac{n(n + 1)(2n + 1)}{6} \]

For \( n = 20 \):

\[ \text{Sum of squares} = \frac{20 \times (20 + 1) \times (2 \times 20 + 1)}{6} \]

Calculating each part:

\[ = \frac{20 \times 21 \times 41}{6} \]

Calculating \( 20 \times 21 \):

\[ 20 \times 21 = 420 \]

Now calculating \( 420 \times 41 \):

\[ 420 \times 41 = 17220 \]

Now divide by \( 6 \):

\[ \frac{17220}{6} = 2870 \]

Thus, the sum of squares of the first 20 natural numbers is \( 2870 \), which confirms that the second statement is correct.

### Conclusion

- The first statement is **incorrect**: The sum of cubes of the first 20 natural numbers is \( 44100 \).
- The second statement is **correct**: The sum of squares of the first 20 natural numbers is \( 2870 \).

### Summary of Statements
- Statement 1: Incorrect
- Statement 2: Correct

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