`(5^2+6^2+7^2+……+10^2)` is equal to

Options

A. a) 330
B. b) 345
C. c) 355
D. d) 360

Answer

To find the sum of the squares from \(5^2\) to \(10^2\), we can use the formula for the sum of squares of the first \(n\) natural numbers, which is given by:

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

### Step 1: Calculate the sum of squares from \(1\) to \(10\)

First, we will calculate the sum of squares from \(1\) to \(10\) using the formula. Here, \(n = 10\).

\[ S_{10} = \frac{10(10 + 1)(2 \cdot 10 + 1)}{6} \]

Calculating this step by step:

1. Calculate \(10 + 1 = 11\)
2. Calculate \(2 \cdot 10 + 1 = 21\)
3. Now substitute these values into the formula:

\[ S_{10} = \frac{10 \cdot 11 \cdot 21}{6} \]

4. Calculate \(10 \cdot 11 = 110\)
5. Calculate \(110 \cdot 21 = 2310\)
6. Finally, divide by \(6\):

\[ S_{10} = \frac{2310}{6} = 385 \]

### Step 2: Calculate the sum of squares from \(1\) to \(4\)

Next, we will calculate the sum of squares from \(1\) to \(4\) using the same formula. Here, \(n = 4\).

\[ S_{4} = \frac{4(4 + 1)(2 \cdot 4 + 1)}{6} \]

Calculating this step by step:

1. Calculate \(4 + 1 = 5\)
2. Calculate \(2 \cdot 4 + 1 = 9\)
3. Now substitute these values into the formula:

\[ S_{4} = \frac{4 \cdot 5 \cdot 9}{6} \]

4. Calculate \(4 \cdot 5 = 20\)
5. Calculate \(20 \cdot 9 = 180\)
6. Finally, divide by \(6\):

\[ S_{4} = \frac{180}{6} = 30 \]

### Step 3: Calculate the sum of squares from \(5\) to \(10\)

Now, we can find the sum of squares from \(5^2\) to \(10^2\) by subtracting the sum of squares from \(1\) to \(4\) from the sum of squares from \(1\) to \(10\):

\[ S_{5 \text{ to } 10} = S_{10} - S_{4} \]

Substituting the values we calculated:

\[ S_{5 \text{ to } 10} = 385 - 30 = 355 \]

### Final Answer

Thus, the sum of squares from \(5^2\) to \(10^2\) is:

\[ \boxed{355} \]