What is the value of `12^(2) - 11^(2) + 14^(2) - 13^(2) + 16^(2) - 15^(2) + 18^(2) - 17^(2) + 20^(2) - 19^(2)` ?

Options

A. 135
B. 198
C. 172
D. 155

Answer

To solve the expression \( 12^2 - 11^2 + 14^2 - 13^2 + 16^2 - 15^2 + 18^2 - 17^2 + 20^2 - 19^2 \), we can use the difference of squares formula, which states that \( a^2 - b^2 = (a - b)(a + b) \).

### Step-by-step Solution:

1. **Identify the pairs**: We can group the terms in pairs:
- \( 12^2 - 11^2 \)
- \( 14^2 - 13^2 \)
- \( 16^2 - 15^2 \)
- \( 18^2 - 17^2 \)
- \( 20^2 - 19^2 \)

2. **Apply the difference of squares formula**:
- For \( 12^2 - 11^2 \):
\[ 12^2 - 11^2 = (12 - 11)(12 + 11) = 1 \cdot 23 = 23 \]
- For \( 14^2 - 13^2 \):
\[ 14^2 - 13^2 = (14 - 13)(14 + 13) = 1 \cdot 27 = 27 \]
- For \( 16^2 - 15^2 \):
\[ 16^2 - 15^2 = (16 - 15)(16 + 15) = 1 \cdot 31 = 31 \]
- For \( 18^2 - 17^2 \):
\[ 18^2 - 17^2 = (18 - 17)(18 + 17) = 1 \cdot 35 = 35 \]
- For \( 20^2 - 19^2 \):
\[ 20^2 - 19^2 = (20 - 19)(20 + 19) = 1 \cdot 39 = 39 \]

3. **Sum the results**:
Now we add all the results together:
\[ 23 + 27 + 31 + 35 + 39 \]

4. **Calculate the total**:
- First, add \( 23 + 27 = 50 \)
- Next, \( 50 + 31 = 81 \)
- Then, \( 81 + 35 = 116 \)
- Finally, \( 116 + 39 = 155 \)

Thus, the value of the expression is \( \boxed{155} \).