Given that `1^(2) + 2^(2) + 3^(2) + …. + n^(2) = (n)/(6) (n + 1) (2n + 1)` , then `10^(2) + 11^(2) + 12^(2) + … + 20^(2)` is equal to

Options

A. 2616
B. 2585
C. 3747
D. 2555

Answer

To solve the problem of finding the sum \(10^2 + 11^2 + 12^2 + \ldots + 20^2\) using the formula for the sum of squares, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula**: We are given the formula for the sum of squares:
\[ 1^2 + 2^2 + 3^2 + \ldots + n^2 = \frac{n}{6} (n + 1)(2n + 1) \]

2. **Calculate the Sum from 1 to 20**: First, we will calculate \(1^2 + 2^2 + \ldots + 20^2\) by substituting \(n = 20\) into the formula:
\[ S_{20} = \frac{20}{6} (20 + 1)(2 \times 20 + 1) \]
Simplifying this:
\[ S_{20} = \frac{20}{6} \times 21 \times 41 \]

3. **Calculate \(S_{20}\)**:
- First, simplify \(\frac{20}{6} = \frac{10}{3}\).
- Now calculate \(21 \times 41\):
\[ 21 \times 41 = 861 \]
- Now substitute back:
\[ S_{20} = \frac{10}{3} \times 861 \]
- Calculate:
\[ S_{20} = 2870 \]

4. **Calculate the Sum from 1 to 9**: Next, we will calculate \(1^2 + 2^2 + \ldots + 9^2\) by substituting \(n = 9\) into the formula:
\[ S_{9} = \frac{9}{6} (9 + 1)(2 \times 9 + 1) \]
Simplifying this:
\[ S_{9} = \frac{9}{6} \times 10 \times 19 \]
- First, simplify \(\frac{9}{6} = \frac{3}{2}\).
- Now calculate \(10 \times 19 = 190\):
\[ S_{9} = \frac{3}{2} \times 190 = 285 \]

5. **Calculate the Required Sum**: Now we can find \(10^2 + 11^2 + \ldots + 20^2\) by subtracting \(S_{9}\) from \(S_{20}\):
\[ 10^2 + 11^2 + \ldots + 20^2 = S_{20} - S_{9} = 2870 - 285 \]
- Calculate:
\[ 2870 - 285 = 2585 \]

### Final Answer:
Thus, the value of \(10^2 + 11^2 + 12^2 + \ldots + 20^2\) is \(2585\).

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