Find `1^(2) + 2^(2) +.................+ 10^(2)`

Explanation

To find the sum of the squares of the first 10 natural numbers, we can use the formula for the sum of squares of the first n natural numbers:

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

where \( n \) is the number of terms. In this case, \( n = 10 \).

### Step-by-step Solution:

1. **Identify the value of \( n \)**:
\[ n = 10 \]

2. **Substitute \( n \) into the formula**:
\[ S = \frac{10(10 + 1)(2 \times 10 + 1)}{6} \]

3. **Calculate \( n + 1 \)**:
\[ n + 1 = 10 + 1 = 11 \]

4. **Calculate \( 2n + 1 \)**:
\[ 2n + 1 = 2 \times 10 + 1 = 20 + 1 = 21 \]

5. **Substitute these values back into the formula**:
\[ S = \frac{10 \times 11 \times 21}{6} \]

6. **Calculate the product in the numerator**:
\[ 10 \times 11 = 110 \]
\[ 110 \times 21 = 2310 \]

7. **Divide by 6**:
\[ S = \frac{2310}{6} \]

8. **Perform the division**:
\[ S = 385 \]

### Final Answer:
The sum of the squares of the first 10 natural numbers is \( 385 \).

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