The sequence $3, 5, 7, 9, \dots$ is an arithmetic progression (AP) where the first term $a_1 = 3$ and the common difference $d = 5 - 3 = 2$.

To find the sum of the first 50 terms of an arithmetic progression, we use the formula for the sum of the first $n$ terms:

$S_n = \frac{n}{2} \times (2a_1 + (n-1)d)$

Here:

• $a_1 = 3$ (the first term),
• $d = 2$ (the common difference),
• $n = 50$ (the number of terms).

Substitute the values into the formula:

$S_{50} = \frac{50}{2} \times (2 \times 3 + (50-1) \times 2)$

Now simplify the expression:

$S_{50} = 25 \times (6 + 49 \times 2)$ $S_{50} = 25 \times (6 + 98)$ $S_{50} = 25 \times 104$ $S_{50} = 2600$

So, the sum of the first 50 terms of the progression is $\mathbf{2600}$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you find the 50th term in this arithmetic progression?
2. What is the formula for the $n$-th term of an arithmetic progression?
3. Can you find the sum of the first 100 terms of this progression?
4. How would the sum change if the common difference were 3 instead of 2?
5. What is the average of the first 50 terms in this sequence?

Tip: Always check if a sequence is arithmetic by verifying that the difference between consecutive terms is constant.