what is the sum of the first 50 terms of progression 3,5,7,9,,,,

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.

What is the sum of the first 50 terms of the progression 3, 5, 7, 9,...?

Calculate the sum of the first 50 terms of the arithmetic progression starting at 3 with a common difference of 2. This detailed solution covers arithmetic series summation, using the formula for the sum of an arithmetic progression. Suitable for students in Grades 9-11.

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