What is the value of `11^(2)+12^(2)+.....+20^(2)`?

Options

A. A)3029
B. B)2485
C. C)2289
D. D)4055

Answer

To find the value of \( 11^2 + 12^2 + 13^2 + \ldots + 20^2 \), we can use the formula for the sum of squares of the first \( n \) natural numbers, which is given by:

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

### Step 1: Calculate the sum of squares from 1 to 20
We first calculate the sum of squares from \( 1^2 \) to \( 20^2 \).

Here, \( n = 20 \):

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

Calculating this step-by-step:

1. \( 20 + 1 = 21 \)
2. \( 2 \times 20 + 1 = 41 \)
3. Now substitute these values into the formula:

\[ \text{Sum from 1 to 20} = \frac{20 \times 21 \times 41}{6} \]

### Step 2: Simplify the calculation
Now, we can simplify:

1. Calculate \( 20 \times 21 = 420 \)
2. Now calculate \( 420 \times 41 \):

\[ 420 \times 41 = 17220 \]

3. Finally, divide by 6:

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

So, the sum of squares from \( 1^2 \) to \( 20^2 \) is \( 2870 \).

### Step 3: Calculate the sum of squares from 1 to 10
Next, we calculate the sum of squares from \( 1^2 \) to \( 10^2 \).

Here, \( n = 10 \):

\[ \text{Sum from 1 to 10} = \frac{10(10 + 1)(2 \times 10 + 1)}{6} \]

Calculating this step-by-step:

1. \( 10 + 1 = 11 \)
2. \( 2 \times 10 + 1 = 21 \)

Now substitute these values into the formula:

\[ \text{Sum from 1 to 10} = \frac{10 \times 11 \times 21}{6} \]

### Step 4: Simplify the calculation
Now, we can simplify:

1. Calculate \( 10 \times 11 = 110 \)
2. Now calculate \( 110 \times 21 \):

\[ 110 \times 21 = 2310 \]

3. Finally, divide by 6:

\[ \frac{2310}{6} = 385 \]

So, the sum of squares from \( 1^2 \) to \( 10^2 \) is \( 385 \).

### Step 5: Calculate the desired sum
Now, we can find the sum from \( 11^2 \) to \( 20^2 \):

\[ 11^2 + 12^2 + \ldots + 20^2 = \text{Sum from 1 to 20} - \text{Sum from 1 to 10} \]

Substituting the values we found:

\[ 11^2 + 12^2 + \ldots + 20^2 = 2870 - 385 = 2485 \]

### Final Answer
Thus, the value of \( 11^2 + 12^2 + \ldots + 20^2 \) is \( 2485 \).