We will use the formula for the $n$-th term of an arithmetic series:

$a_n = a_1 + (n - 1) d$

where:

• $a_n$ is the $n$-th term,
• $a_1$ is the first term,
• $d$ is the common difference, and
• $n$ is the number of terms.

To find the sum of the arithmetic series, we also use the sum formula:

$S_n = \frac{n}{2} \cdot (a_1 + a_n)$

where:

• $S_n$ is the sum of the first $n$ terms,
• $a_1$ is the first term,
• $a_n$ is the last term,
• $n$ is the number of terms.

1. $1 + 2 + 3 + \cdots + 100$

For this arithmetic series:

• First term, $a_1 = 1$,
• Last term, $a_n = 100$,
• Common difference, $d = 1$.

Now, the number of terms $n$ is 100. Using the sum formula:

$S_n = \frac{n}{2} \cdot (a_1 + a_n) = \frac{100}{2} \cdot (1 + 100) = 50 \cdot 101 = 5050$

So, the sum is 5050.

2. $500 + 490 + 480 + \cdots + 300$

For this arithmetic series:

• First term, $a_1 = 500$,
• Last term, $a_n = 300$,
• Common difference, $d = -10$.

Now, we find the number of terms $n$. Using the formula for the $n$-th term:

$a_n = a_1 + (n - 1)d$ $300 = 500 + (n - 1)(-10)$ $300 = 500 - 10(n - 1)$ $300 - 500 = -10(n - 1)$ $-200 = -10(n - 1)$ $n - 1 = 20 \quad \Rightarrow \quad n = 21$

Now, we can calculate the sum:

$S_n = \frac{n}{2} \cdot (a_1 + a_n) = \frac{21}{2} \cdot (500 + 300) = 10.5 \cdot 800 = 8400$

So, the sum is 8400.

3. $2001 + 2002 + 2003 + \cdots + 5000$

For this arithmetic series:

• First term, $a_1 = 2001$,
• Last term, $a_n = 5000$,
• Common difference, $d = 1$.

Now, we find the number of terms $n$. Using the formula for the $n$-th term:

$a_n = a_1 + (n - 1)d$ $5000 = 2001 + (n - 1)(1)$ $5000 = 2001 + n - 1$ $5000 = 2000 + n$ $n = 3000$

Now, we can calculate the sum:

$S_n = \frac{n}{2} \cdot (a_1 + a_n) = \frac{3000}{2} \cdot (2001 + 5000) = 1500 \cdot 7001 = 10501500$

So, the sum is 10,501,500.

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Would you like more details or have any questions? Here are some related problems you can try:

1. What is the sum of the series $1 + 4 + 7 + \cdots + 100$?
2. Evaluate the sum of $100 + 90 + 80 + \cdots + 10$.
3. What is the sum of the first 50 terms of the series $3 + 6 + 9 + \cdots$?
4. Find the sum of $1000 + 995 + 990 + \cdots + 500$.
5. Calculate the sum of $10 + 15 + 20 + \cdots + 1000$.

Tip: For any arithmetic sequence, always check the common difference $d$ before applying the formulas.