Introduction:
In this problem, we are given a sequence of colored beads placed in a specific order. The pattern of the sequence is as follows: 1 red bead, 1 green bead; then 2 red beads, 2 green beads; then 3 red beads, 3 green beads, and so on. We are required to determine the number of red beads in the first 100 beads of the sequence.

Approach:
To solve this problem, we can analyze the pattern and observe that the number of beads in each group (red and green) is increasing by 1 in each step. This implies that the number of beads in each group follows an arithmetic progression.

Step 1: Analyzing the pattern
Let's break down the sequence into groups and count the number of red beads in each group:

Group 1: 1 red, 1 green (Total beads = 2, Red beads = 1)
Group 2: 2 red, 2 green (Total beads = 4, Red beads = 2)
Group 3: 3 red, 3 green (Total beads = 6, Red beads = 3)
...

Step 2: Finding the number of beads in each group
We can observe that the number of beads in each group is equal to the group number multiplied by 2. For example, in Group 1, the number of beads is 1+1 = 2; in Group 2, the number of beads is 2+2 = 4; and so on.

So, in general, the number of beads in each group can be calculated using the formula: Total beads = Group number * 2.

Step 3: Calculating the number of red beads
To determine the number of red beads in each group, we can divide the total number of beads in each group by 2 (since the number of red and green beads are equal in each group).

So, the number of red beads in each group is equal to the group number.

Step 4: Calculating the number of red beads in the first 100 beads
To find the number of red beads in the first 100 beads, we need to sum up the number of red beads in each group from Group 1 to Group 100.

Using the formula for the sum of an arithmetic progression, we can calculate the sum of the first 100 natural numbers as follows:

Sum = (n/2) * (first term + last term), where n is the number of terms.

In our case, the first term is 1 and the last term is 100.

Substituting the values, we get:

Sum = (100/2) * (1 + 100) = 50 * 101 = 5050.

Therefore, the number of red beads in the first 100 beads is 5050.