`60^(2)-59^(2)+58^(2)-57^(2)+ cdots +2^(2)-1^(2)` =

Options

A. 1830
B. 3180
C. 1380
D. none of these

Answer

To solve the expression \( 60^2 - 59^2 + 58^2 - 57^2 + \cdots + 2^2 - 1^2 \), we can use the difference of squares and rearrange the terms. Let's go through the steps systematically.

### Step 1: Group the Terms
We can group the terms in pairs:
\[ (60^2 - 59^2) + (58^2 - 57^2) + \cdots + (2^2 - 1^2) \]

### Step 2: Apply the Difference of Squares Formula
Recall that \( a^2 - b^2 = (a - b)(a + b) \). Applying this to each pair:
\[ 60^2 - 59^2 = (60 - 59)(60 + 59) = 1 \cdot 119 = 119 \]
\[ 58^2 - 57^2 = (58 - 57)(58 + 57) = 1 \cdot 115 = 115 \]
Continuing this pattern, we find:
\[ 56^2 - 55^2 = 1 \cdot 111 = 111 \]
\[ 54^2 - 53^2 = 1 \cdot 107 = 107 \]
And so on, down to:
\[ 2^2 - 1^2 = (2 - 1)(2 + 1) = 1 \cdot 3 = 3 \]

### Step 3: Write the Series of Results
The results of the pairs can be expressed as:
\[ 119 + 115 + 111 + 107 + \cdots + 3 \]

### Step 4: Identify the Sequence
This is an arithmetic series where:
- The first term \( a = 119 \)
- The last term \( l = 3 \)
- The common difference \( d = -4 \)

### Step 5: Find the Number of Terms
To find the number of terms \( n \):
\[ l = a + (n-1)d \]
Substituting the known values:
\[ 3 = 119 + (n-1)(-4) \]
\[ 3 - 119 = -4(n - 1) \]
\[ -116 = -4(n - 1) \]
\[ n - 1 = 29 \quad \Rightarrow \quad n = 30 \]

### Step 6: Calculate the Sum of the Series
The sum \( S_n \) of an arithmetic series can be calculated using the formula:
\[ S_n = \frac{n}{2} (a + l) \]
Substituting the values:
\[ S_{30} = \frac{30}{2} (119 + 3) = 15 \cdot 122 = 1830 \]

### Final Answer
Thus, the value of the expression \( 60^2 - 59^2 + 58^2 - 57^2 + \cdots + 2^2 - 1^2 \) is:
\[ \boxed{1830} \]