The given series is an arithmetic progression (AP) where the first term $a = 1$, the common difference $d = 2$, and the last term $l = 1001$.

The sum of an arithmetic progression can be calculated using the formula:

$S_n = \frac{n}{2} \cdot (a + l)$

Where:

• $S_n$ is the sum of the first $n$ terms,
• $n$ is the number of terms,
• $a$ is the first term, and
• $l$ is the last term.

Step 1: Find the number of terms, $n$.

The general formula for the $n$-th term of an arithmetic progression is:

$l = a + (n - 1) \cdot d$

Substituting the known values:

$1001 = 1 + (n - 1) \cdot 2$

Simplifying the equation:

$1001 - 1 = (n - 1) \cdot 2$ $1000 = (n - 1) \cdot 2$ $n - 1 = 500$ $n = 501$

So, there are 501 terms in the series.

Step 2: Calculate the sum.

Now, we can use the sum formula:

$S_{501} = \frac{501}{2} \cdot (1 + 1001)$ $S_{501} = \frac{501}{2} \cdot 1002$ $S_{501} = 501 \cdot 501$ $S_{501} = 251001$

So, the sum of the series $1 + 3 + 5 + 7 + \dots + 1001$ is 251,001.

Let me know if you'd like any further explanations or details!

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Here are some related questions you might find interesting:

1. What is the formula for the sum of an arithmetic series?
2. How do you calculate the common difference of an arithmetic sequence?
3. How can we use the formula for the $n$-th term of an arithmetic progression?
4. What happens if the common difference is negative in an arithmetic sequence?
5. How do you handle series with a different starting number, such as starting at 3 instead of 1?

Tip: Always double-check the number of terms in your sequence before applying the sum formula!