`sum_(r=0)^(2006)(-1)^(r)(a+rd)=`

Options

A. a+2007d
B. a+1004d
C. a+1003d
D. none of these

Answer

To solve the summation \( S = \sum_{r=0}^{2006} (-1)^r (a + rd) \), we can break it down step by step.

### Step 1: Write out the summation
We start by expanding the summation:
\[ S = (-1)^0 (a + 0d) + (-1)^1 (a + 1d) + (-1)^2 (a + 2d) + \ldots + (-1)^{2006} (a + 2006d) \]
This simplifies to:
\[ S = a - (a + d) + (a + 2d) - (a + 3d) + \ldots + (a + 2006d) \]

### Step 2: Group the terms
Next, we can group the terms based on whether \( r \) is even or odd:
- For even \( r \): \( a + 0d, a + 2d, a + 4d, \ldots, a + 2006d \)
- For odd \( r \): \( -(a + 1d), -(a + 3d), -(a + 5d), \ldots, -(a + 2005d) \)

### Step 3: Count the number of terms
There are a total of 2007 terms (from \( r = 0 \) to \( r = 2006 \)):
- There are 1004 even terms (0, 2, 4, ..., 2006).
- There are 1003 odd terms (1, 3, 5, ..., 2005).

### Step 4: Calculate contributions from even and odd terms
The contribution from the even terms:
\[ \text{Even terms} = 1004a + (0 + 2 + 4 + \ldots + 2006)d \]
The sum of the even integers from 0 to 2006 can be calculated using the formula for the sum of an arithmetic series:
\[ \text{Sum} = \frac{n}{2} \times (\text{first term} + \text{last term}) = \frac{1004}{2} \times (0 + 2006) = 502 \times 2006 = 1003 \times 2006 \]
So, the contribution from the even terms is:
\[ 1004a + 1003 \times 2006d \]

The contribution from the odd terms:
\[ \text{Odd terms} = -1003a - (1 + 3 + 5 + \ldots + 2005)d \]
The sum of the odd integers from 1 to 2005 can also be calculated:
\[ \text{Sum} = \frac{n}{2} \times (\text{first term} + \text{last term}) = \frac{1003}{2} \times (1 + 2005) = 501.5 \times 2006 = 1003 \times 1003 \]
So, the contribution from the odd terms is:
\[ -1003a - 1003 \times 1003d \]

### Step 5: Combine contributions
Now, we combine the contributions from even and odd terms:
\[ S = (1004a + 1003 \times 2006d) + (-1003a - 1003 \times 1003d) \]
This simplifies to:
\[ S = (1004 - 1003)a + (1003 \times 2006 - 1003 \times 1003)d \]
\[ S = a + 1003(2006 - 1003)d \]

### Final Step: Simplify the expression
Thus, the final answer is:
\[ S = a + 1003d \]