Consider the set of intergers ` P = { 1,2,3,4…..}`. Let us define A as the average of the odd intergers in P and B as the average of the even integers in P. What is the value of A - B ?

Options

A. `-1`
B. 0
C. 1
D. 2

Answer

To solve the problem, we need to find the averages of the odd and even integers in the set of positive integers \( P = \{ 1, 2, 3, 4, \ldots \} \) and then calculate \( A - B \), where \( A \) is the average of the odd integers and \( B \) is the average of the even integers.

### Step 1: Define the odd and even integers in the set \( P \)

The odd integers in \( P \) are:
\[ 1, 3, 5, 7, 9, \ldots \]

The even integers in \( P \) are:
\[ 2, 4, 6, 8, 10, \ldots \]

### Step 2: Calculate the average of the odd integers \( A \)

The average of the first \( n \) odd integers can be calculated using the formula:
\[ A = \frac{\text{Sum of the first } n \text{ odd integers}}{n} \]

The sum of the first \( n \) odd integers is given by:
\[ \text{Sum} = n^2 \]

Thus, the average \( A \) becomes:
\[ A = \frac{n^2}{n} = n \]

### Step 3: Calculate the average of the even integers \( B \)

The average of the first \( n \) even integers can be calculated using the formula:
\[ B = \frac{\text{Sum of the first } n \text{ even integers}}{n} \]

The sum of the first \( n \) even integers is given by:
\[ \text{Sum} = n(n + 1) \]

Thus, the average \( B \) becomes:
\[ B = \frac{n(n + 1)}{n} = n + 1 \]

### Step 4: Calculate \( A - B \)

Now we need to find \( A - B \):
\[ A - B = n - (n + 1) = n - n - 1 = -1 \]

### Final Answer

The value of \( A - B \) is:
\[ \boxed{-1} \]

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