To evaluate the summation $S = \sum_{k=1}^{40} (25 - 2k)$, we proceed step by step:

Step 1: Break the summation into two separate sums

The given summation can be split as follows:

$S = \sum_{k=1}^{40} (25 - 2k) = \sum_{k=1}^{40} 25 - \sum_{k=1}^{40} 2k$

Step 2: Solve each part

1. The first term $\sum_{k=1}^{40} 25$:
   Since $25$ is constant, it repeats for $40$ terms:

   $\sum_{k=1}^{40} 25 = 25 \cdot 40 = 1000$

2. The second term $\sum_{k=1}^{40} 2k$:
   Factor out $2$ from the summation:

   $\sum_{k=1}^{40} 2k = 2 \cdot \sum_{k=1}^{40} k$

   The sum of the first $n$ natural numbers is given by:

   $\sum_{k=1}^n k = \frac{n(n+1)}{2}$

   Substituting $n = 40$:

   $\sum_{k=1}^{40} k = \frac{40 \cdot (40 + 1)}{2} = \frac{40 \cdot 41}{2} = 820$

   Therefore:

   $\sum_{k=1}^{40} 2k = 2 \cdot 820 = 1640$

Step 3: Combine the results

Now substitute the results into the original equation:

$S = \sum_{k=1}^{40} 25 - \sum_{k=1}^{40} 2k$

$S = 1000 - 1640$

$S = -640$

Final Answer:

$\boxed{-640}$

Let me know if you need further clarifications or a deeper explanation!

Related Questions:

1. What is the general formula for the sum of an arithmetic series?
2. How do you evaluate a summation with a linear expression like $a + bk$?
3. How can the sum of squares of the first $n$ natural numbers be calculated?
4. What happens when the summation index starts at a value other than 1?
5. How do you handle summations involving multiple variables?

Tip: Always split a summation with multiple terms into smaller, manageable parts using properties of summation!