Find the sum of all odd integers from 1 to 201.

Explanation

To find the sum of all odd integers from 1 to 201, we can follow these steps:

### Step 1: Identify the sequence of odd integers
The odd integers from 1 to 201 form an arithmetic progression (AP) where:
- The first term (A) is 1.
- The common difference (D) is 2.
- The last term (L) is 201.

### Step 2: Find the number of terms (n) in the sequence
We can use the formula for the nth term of an arithmetic progression:
\[ T_n = A + (n - 1) \cdot D \]
Setting \( T_n = 201 \):
\[ 201 = 1 + (n - 1) \cdot 2 \]
Now, simplify the equation:
\[ 201 - 1 = (n - 1) \cdot 2 \]
\[ 200 = (n - 1) \cdot 2 \]
Dividing both sides by 2:
\[ 100 = n - 1 \]
Adding 1 to both sides gives:
\[ n = 101 \]

### Step 3: Calculate the sum of the first n terms (S_n)
The sum of the first n terms of an arithmetic progression can be calculated using the formula:
\[ S_n = \frac{n}{2} \cdot (A + L) \]
Substituting the values we found:
\[ S_{101} = \frac{101}{2} \cdot (1 + 201) \]
\[ S_{101} = \frac{101}{2} \cdot 202 \]
Calculating this gives:
\[ S_{101} = 101 \cdot 101 = 10201 \]

### Final Answer
The sum of all odd integers from 1 to 201 is **10201**.

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