`11^(2)+12^(2)+13^(2)+ cdots +32^(2)`=

Options

A. 11550
B. 11055
C. 55011
D. none of these

Answer

To solve the problem of finding the sum of the squares from \(11^2\) to \(32^2\), we can use the formula for the sum of squares of the first \(n\) natural numbers. The formula for the sum of squares of the first \(n\) natural numbers is given by:

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

### Step-by-Step Solution:

1. **Calculate the sum of squares from \(1^2\) to \(32^2\)**:
- We need to find \(S_{32}\):
\[ S_{32} = \frac{32(32 + 1)(2 \cdot 32 + 1)}{6} \]
- Calculate:
\[ S_{32} = \frac{32 \cdot 33 \cdot 65}{6} \]
- First, calculate \(32 \cdot 33 = 1056\).
- Then, calculate \(1056 \cdot 65 = 68640\).
- Finally, divide by \(6\):
\[ S_{32} = \frac{68640}{6} = 11440 \]

2. **Calculate the sum of squares from \(1^2\) to \(10^2\)**:
- Now we need to find \(S_{10}\):
\[ S_{10} = \frac{10(10 + 1)(2 \cdot 10 + 1)}{6} \]
- Calculate:
\[ S_{10} = \frac{10 \cdot 11 \cdot 21}{6} \]
- First, calculate \(10 \cdot 11 = 110\).
- Then, calculate \(110 \cdot 21 = 2310\).
- Finally, divide by \(6\):
\[ S_{10} = \frac{2310}{6} = 385 \]

3. **Find the sum from \(11^2\) to \(32^2\)**:
- Now, we subtract \(S_{10}\) from \(S_{32}\):
\[ S_{11 \text{ to } 32} = S_{32} - S_{10} = 11440 - 385 = 11055 \]

### Final Answer:
The sum \(11^2 + 12^2 + 13^2 + \cdots + 32^2\) is \(11055\).