`1^2-2^2+3^2-4^2+…….-10^2` is equal to

Options

A. 45
B. `-45`
C. `-54`
D. `-55`

Answer

To solve the expression \(1^2 - 2^2 + 3^2 - 4^2 + \ldots - 10^2\), we can group the terms in pairs and use the difference of squares formula.

### Step-by-step Solution:

1. **Group the terms**:
We can group the terms in pairs as follows:
\[ (1^2 - 2^2) + (3^2 - 4^2) + (5^2 - 6^2) + (7^2 - 8^2) + (9^2 - 10^2) \]

2. **Apply the difference of squares formula**:
Recall that \(a^2 - b^2 = (a + b)(a - b)\). We can apply this to each pair:
- For \(1^2 - 2^2\):
\[ 1^2 - 2^2 = (1 + 2)(1 - 2) = 3 \cdot (-1) = -3 \]
- For \(3^2 - 4^2\):
\[ 3^2 - 4^2 = (3 + 4)(3 - 4) = 7 \cdot (-1) = -7 \]
- For \(5^2 - 6^2\):
\[ 5^2 - 6^2 = (5 + 6)(5 - 6) = 11 \cdot (-1) = -11 \]
- For \(7^2 - 8^2\):
\[ 7^2 - 8^2 = (7 + 8)(7 - 8) = 15 \cdot (-1) = -15 \]
- For \(9^2 - 10^2\):
\[ 9^2 - 10^2 = (9 + 10)(9 - 10) = 19 \cdot (-1) = -19 \]

3. **Sum the results**:
Now we sum all the results from the pairs:
\[ -3 - 7 - 11 - 15 - 19 \]

4. **Calculate the total**:
Let's add these values together:
- First, add \(-3\) and \(-7\):
\[ -3 - 7 = -10 \]
- Next, add \(-11\):
\[ -10 - 11 = -21 \]
- Then, add \(-15\):
\[ -21 - 15 = -36 \]
- Finally, add \(-19\):
\[ -36 - 19 = -55 \]

Thus, the final result is:
\[ \boxed{-55} \]