91Ó°ÊÓ

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference. For example, in the sequence 1, 3, 5, 7, 9, ..., each term increases by 2, making 2 the common difference.

In this exercise, the odd numbers from 1 to 99 form an arithmetic sequence with the first term (a) as 1 and the common difference (d) as 2. To find the number of terms (n) in this sequence, we use the general formula: \(a_n = a + (n-1)d\). Substituting the known values: \[99 = 1 + (n-1)*2\]. Solving for \(n\), we get 50. This means there are 50 odd numbers between 1 and 100.

Probability theory deals with the study of uncertainty and the likelihood of different outcomes. In this context, it helps us determine the probability of selecting a random element from a given set.

To calculate the probability, we use the formula \(P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\). Here, the event (E) is choosing an odd number from the first 100 positive integers. The number of favorable outcomes is 50 (since there are 50 odd numbers), and the total number of possible outcomes is 100. Substituting these into the formula gives us \(P(\text{odd}) = \frac{50}{100} = \frac{1}{2}\). This tells us that there is a 50% chance of selecting an odd number.

Positive integers are all the whole numbers greater than zero. They are often denoted by the set notation \(\mathbb{Z}^+\) or simply \(1, 2, 3, 4, ...\).

In this exercise, we are focusing on the first 100 positive integers, which means the set \[1, 2, 3, ..., 100\]. Identifying the total number of integers within this range helps us determine the sample space for our probability calculation. The analysis shows that out of these 100 integers, exactly half (50) are odd. This balanced division of odd and even numbers nicely simplifies the probability calculation.