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Answer:
To find the sum of all odd numbers from 1 to 101, you can use the formula for the sum of an arithmetic series. Here's a step-by-step approach:
1. **Identify the Series**: The series of odd numbers from 1 to 101 is: \( 1, 3, 5, \ldots, 101 \).
2. **Find the Number of Terms**: The common difference \( d \) is 2. To find the number of terms (\( n \)), use the formula for the nth term of an arithmetic series:
\[
a_n = a + (n - 1)d
\]
where \( a \) is the first term (1), \( d \) is the common difference (2), and \( a_n \) is the last term (101). Setting up the equation:
\[
101 = 1 + (n - 1) \cdot 2
\]
Solving for \( n \):
\[
101 = 1 + 2n - 2
\]
\[
101 = 2n - 1
\]
\[
102 = 2n
\]
\[
n = 51
\]
3. **Calculate the Sum**: The sum \( S_n \) of an arithmetic series is given by:
\[
S_n = \frac{n}{2} \cdot (a + l)
\]
where \( a \) is the first term, \( l \) is the last term, and \( n \) is the number of terms. Substituting the values:
\[
S_{51} = \frac{51}{2} \cdot (1 + 101)
\]
\[
S_{51} = \frac{51}{2} \cdot 102
\]
\[
S_{51} = 51 \cdot 51
\]
\[
S_{51} = 2601
\]
So, the sum of all odd numbers from 1 to 101 is \( 2601 \).