The value of `(1^2 - 2^2 + 3^2 - 4^2 +5^2 -6^2 +.....+ 9^2 - 10^2)` is

Options

A. 110
B. `-110`
C. 55
D. `-55`

Answer

To solve the expression \( (1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + 7^2 - 8^2 + 9^2 - 10^2) \), we will use the difference of squares formula, which states that \( a^2 - b^2 = (a - b)(a + b) \).

### Step-by-step Solution:

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

2. **Apply the difference of squares**:
- For \( 1^2 - 2^2 \):
\[ 1^2 - 2^2 = (1 - 2)(1 + 2) = (-1)(3) = -3 \]
- For \( 3^2 - 4^2 \):
\[ 3^2 - 4^2 = (3 - 4)(3 + 4) = (-1)(7) = -7 \]
- For \( 5^2 - 6^2 \):
\[ 5^2 - 6^2 = (5 - 6)(5 + 6) = (-1)(11) = -11 \]
- For \( 7^2 - 8^2 \):
\[ 7^2 - 8^2 = (7 - 8)(7 + 8) = (-1)(15) = -15 \]
- For \( 9^2 - 10^2 \):
\[ 9^2 - 10^2 = (9 - 10)(9 + 10) = (-1)(19) = -19 \]

3. **Sum the results**:
Now we add all the results from the pairs:
\[ -3 + (-7) + (-11) + (-15) + (-19) \]
- First, add \( -3 \) and \( -7 \):
\[ -3 - 7 = -10 \]
- Next, add \( -11 \):
\[ -10 - 11 = -21 \]
- Then, add \( -15 \):
\[ -21 - 15 = -36 \]
- Finally, add \( -19 \):
\[ -36 - 19 = -55 \]

4. **Final Result**:
The value of the expression \( (1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + 7^2 - 8^2 + 9^2 - 10^2) \) is:
\[ \boxed{-55} \]