The problem asks us to find the sum of all positive odd integers less than 50.

1. Problem Description

The problem asks us to find the sum of all positive odd integers less than

5

0.

2. Solution Steps

We want to find the sum of the arithmetic series $1 + 3 + 5 + \dots + 49$.

First, we need to determine the number of terms in this series. The general form of an odd integer is $2n-1$, where $n$ is an integer.

We want to find the value of $n$ for which $2n-1 = 49$.

$2n - 1 = 49$

$2n = 50$

$n = 25$

So there are 25 terms in this series.

The sum of an arithmetic series is given by the formula:

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

where $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the last term. In this case, $n = 25$, $a_1 = 1$, and $a_n = 49$.

Therefore, the sum is:

$S_{25} = \frac{25}{2}(1 + 49)$

$S_{25} = \frac{25}{2}(50)$

$S_{25} = 25 \cdot 25$

$S_{25} = 625$

3. Final Answer

The sum of positive odd integers less than 50 is 625.