`8^(2)+9^(2)+10^(2)+ cdots +22^(2)`=

Options

A. 3656
B. 3655
C. 3654
D. none of these

Answer

To find the sum of the series \(8^2 + 9^2 + 10^2 + \cdots + 22^2\), we can use the formula for the sum of squares of the first \(n\) natural numbers. The formula for the sum of squares is:

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

### Step 1: Calculate the sum of squares from \(1^2\) to \(22^2\)

First, we will calculate the sum of squares from \(1^2\) to \(22^2\):

- Here, \(n = 22\).

Using the formula:

\[ S_{22} = \frac{22(22 + 1)(2 \cdot 22 + 1)}{6} \]

Calculating each part:

- \(22 + 1 = 23\)
- \(2 \cdot 22 + 1 = 45\)

Now substituting these values into the formula:

\[ S_{22} = \frac{22 \cdot 23 \cdot 45}{6} \]

Calculating this step-by-step:

1. \(22 \cdot 23 = 506\)
2. \(506 \cdot 45 = 22770\)
3. Finally, divide by \(6\):

\[ S_{22} = \frac{22770}{6} = 3795 \]

### Step 2: Calculate the sum of squares from \(1^2\) to \(7^2\)

Next, we will calculate the sum of squares from \(1^2\) to \(7^2\):

- Here, \(n = 7\).

Using the formula:

\[ S_{7} = \frac{7(7 + 1)(2 \cdot 7 + 1)}{6} \]

Calculating each part:

- \(7 + 1 = 8\)
- \(2 \cdot 7 + 1 = 15\)

Now substituting these values into the formula:

\[ S_{7} = \frac{7 \cdot 8 \cdot 15}{6} \]

Calculating this step-by-step:

1. \(7 \cdot 8 = 56\)
2. \(56 \cdot 15 = 840\)
3. Finally, divide by \(6\):

\[ S_{7} = \frac{840}{6} = 140 \]

### Step 3: Calculate the sum of squares from \(8^2\) to \(22^2\)

Now, we can find the sum of squares from \(8^2\) to \(22^2\) by subtracting \(S_{7}\) from \(S_{22}\):

\[ S_{8 \text{ to } 22} = S_{22} - S_{7} \]

Substituting the values we calculated:

\[ S_{8 \text{ to } 22} = 3795 - 140 = 3655 \]

### Final Answer

Thus, the sum \(8^2 + 9^2 + 10^2 + \cdots + 22^2\) is:

\[ \boxed{3655} \]