If the sum of m consecutive odd integers is `m^(4)` , then the first integer is

Options

A. `m^(3)+m+1`
B. `m^(3)+m-1`
C. `m^(3)-m-1`
D. `m^(3)-m+1`

Explanation

To solve the problem of finding the first integer when the sum of \( m \) consecutive odd integers is \( m^4 \), we can follow these steps:

### Step 1: Define the first integer
Let the first odd integer be \( A \). The next consecutive odd integers can be expressed as:
- First integer: \( A \)
- Second integer: \( A + 2 \)
- Third integer: \( A + 4 \)
- ...
- \( m \)-th integer: \( A + 2(m - 1) \)

### Step 2: Write the sum of the integers
The sum of these \( m \) consecutive odd integers can be represented as:
\[ \text{Sum} = A + (A + 2) + (A + 4) + \ldots + (A + 2(m - 1)) \]
This can be simplified to:
\[ \text{Sum} = mA + 2(0 + 1 + 2 + \ldots + (m - 1)) \]

### Step 3: Use the formula for the sum of the first \( n \) integers
The sum of the first \( n \) integers is given by:
\[ 0 + 1 + 2 + \ldots + (m - 1) = \frac{(m - 1)m}{2} \]
Thus, substituting this into our sum gives:
\[ \text{Sum} = mA + 2 \cdot \frac{(m - 1)m}{2} = mA + (m - 1)m \]
This simplifies to:
\[ \text{Sum} = mA + m^2 - m \]

### Step 4: Set the sum equal to \( m^4 \)
According to the problem, this sum is equal to \( m^4 \):
\[ mA + m^2 - m = m^4 \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ mA = m^4 - m^2 + m \]
Dividing both sides by \( m \) (assuming \( m \neq 0 \)):
\[ A = m^3 - m + 1 \]

### Final Answer
Thus, the first odd integer \( A \) in terms of \( m \) is:
\[ A = m^3 - m + 1 \]