Find the sum of the squares of first 50 natural numbers

Options

A. 463
B. 466
C. 456
D. 42925

Answer

To find the sum of the squares of the first 50 natural numbers, we can use the formula:

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

where \( S_n \) is the sum of the squares of the first \( n \) natural numbers.

### Step 1: Identify \( n \)
In this case, \( n = 50 \).

### Step 2: Substitute \( n \) into the formula
We will substitute \( n = 50 \) into the formula:

\[ S_{50} = \frac{50(50 + 1)(2 \times 50 + 1)}{6} \]

### Step 3: Calculate \( 50 + 1 \) and \( 2 \times 50 + 1 \)
First, calculate \( 50 + 1 \):

\[ 50 + 1 = 51 \]

Next, calculate \( 2 \times 50 + 1 \):

\[ 2 \times 50 + 1 = 100 + 1 = 101 \]

### Step 4: Substitute these values back into the formula
Now, substitute these values back into the formula:

\[ S_{50} = \frac{50 \times 51 \times 101}{6} \]

### Step 5: Calculate the product \( 50 \times 51 \times 101 \)
First, calculate \( 50 \times 51 \):

\[ 50 \times 51 = 2550 \]

Now, multiply this result by \( 101 \):

\[ 2550 \times 101 = 257550 \]

### Step 6: Divide the product by 6
Now, divide \( 257550 \) by \( 6 \):

\[ S_{50} = \frac{257550}{6} = 42925 \]

### Final Answer
Thus, the sum of the squares of the first 50 natural numbers is:

\[ \boxed{42925} \]