Find the sum of odd integers from 1 to 2001.

Explanation

To find the sum of all odd integers from 1 to 2001, we can follow these steps:

### Step 1: Identify the sequence of odd integers
The odd integers from 1 to 2001 can be expressed as:
1, 3, 5, 7, ..., 2001

### Step 2: Determine the first term (A) and the last term (L)
- The first term \( A = 1 \)
- The last term \( L = 2001 \)

### Step 3: Identify the common difference (d)
The common difference \( d \) between consecutive odd integers is:
- \( d = 3 - 1 = 2 \)

### Step 4: Find the number of terms (n)
To find the number of terms in this sequence, we can use the formula for the nth term of an arithmetic progression (AP):
\[ A_n = A + (n - 1)d \]
Setting \( A_n = 2001 \):
\[ 2001 = 1 + (n - 1) \cdot 2 \]
Subtracting 1 from both sides gives:
\[ 2000 = (n - 1) \cdot 2 \]
Dividing both sides by 2:
\[ 1000 = n - 1 \]
Adding 1 to both sides gives:
\[ n = 1001 \]

### Step 5: Calculate the sum of the first n terms (S_n)
The formula for the sum of the first n terms of an arithmetic series is:
\[ S_n = \frac{n}{2} \cdot (A + L) \]
Substituting the values we found:
\[ S_{1001} = \frac{1001}{2} \cdot (1 + 2001) \]
Calculating the sum inside the parentheses:
\[ S_{1001} = \frac{1001}{2} \cdot 2002 \]
Now, multiplying:
\[ S_{1001} = 1001 \cdot 1001 \]
Calculating \( 1001^2 \):
\[ S_{1001} = 1002001 \]

### Final Answer
The sum of all odd integers from 1 to 2001 is \( 1002001 \).

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