Let `S` be the set of all value of `a_(1)` for which the mean deviation about the mean deviation about the mean of `100` consecutive positive integers `a_(1),a_(2),a_(3) . . . a_(100)` is 25.Then `S` is

Options

A. `{99}`
B. `phi`
C. `N`
D. `{9}`

Answer

To solve the problem, we need to find the set \( S \) of all values of \( a_1 \) for which the mean deviation about the mean of 100 consecutive positive integers \( a_1, a_2, a_3, \ldots, a_{100} \) is 25.

### Step 1: Define the integers
The 100 consecutive integers can be expressed as:
\[ a_1, a_1 + 1, a_1 + 2, \ldots, a_1 + 99 \]

### Step 2: Calculate the mean
The mean \( \bar{x} \) of these integers is given by:
\[ \bar{x} = \frac{a_1 + (a_1 + 1) + (a_1 + 2) + \ldots + (a_1 + 99)}{100} \]
This simplifies to:
\[ \bar{x} = \frac{100a_1 + (0 + 1 + 2 + \ldots + 99)}{100} \]
Using the formula for the sum of the first \( n \) integers, \( \frac{n(n + 1)}{2} \), we find:
\[ 0 + 1 + 2 + \ldots + 99 = \frac{99 \times 100}{2} = 4950 \]
Thus, the mean becomes:
\[ \bar{x} = \frac{100a_1 + 4950}{100} = a_1 + 49.5 \]

### Step 3: Calculate the mean deviation
The mean deviation \( D \) about the mean is defined as:
\[ D = \frac{\sum_{i=1}^{100} |x_i - \bar{x}|}{100} \]
Where \( x_i \) are the integers \( a_1, a_1 + 1, \ldots, a_1 + 99 \).

Calculating \( |x_i - \bar{x}| \):
- For \( x_1 = a_1 \):
\[ |a_1 - (a_1 + 49.5)| = | -49.5 | = 49.5 \]
- For \( x_2 = a_1 + 1 \):
\[ |(a_1 + 1) - (a_1 + 49.5)| = |1 - 49.5| = 48.5 \]
- Continuing this pattern, we find:
\[ |x_i - \bar{x}| = |(a_1 + (i - 1)) - (a_1 + 49.5)| = |(i - 1) - 49.5| = |i - 50.5| \]

### Step 4: Calculate the total deviation
The values of \( |i - 50.5| \) for \( i = 1, 2, \ldots, 100 \) will yield:
- For \( i = 1 \) to \( 50 \), the values are \( 49.5, 48.5, \ldots, 0.5 \).
- For \( i = 51 \) to \( 100 \), the values are \( 0.5, 1.5, \ldots, 49.5 \).

Thus, the total deviation is:
\[ \sum_{i=1}^{100} |i - 50.5| = 2 \times (49.5 + 48.5 + \ldots + 0.5) \]
This is twice the sum of the first 49.5 integers:
\[ = 2 \times \frac{49.5 \times 50}{2} = 49.5 \times 50 = 2475 \]

### Step 5: Mean deviation
Now, we calculate the mean deviation:
\[ D = \frac{2475}{100} = 24.75 \]

### Step 6: Set the mean deviation equal to 25
We set the mean deviation equal to 25:
\[ \frac{2475 + k}{100} = 25 \]
Solving for \( k \):
\[ 2475 + k = 2500 \implies k = 25 \]

### Step 7: Conclusion
The mean deviation can be adjusted by changing \( a_1 \). Since \( a_1 \) can take any positive integer value, the set \( S \) is all positive integers.

Thus, the final answer is:
\[ S = \{ a_1 \in \mathbb{N} \} \]