To find the sum of all odd numbers between 1 and 100 that are divisible by 3, we can follow these steps:
### Step 1: Identify the odd numbers between 1 and 100 that are divisible by 3.
The odd numbers between 1 and 100 that are divisible by 3 are:
- 3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93, 99.
### Step 2: Verify that these numbers form an arithmetic progression (AP).
We can see that the first term \( a = 3 \) and the common difference \( d = 6 \) (since each term increases by 6).
### Step 3: Determine the number of terms in the series.
To find the number of terms \( n \), we can use the formula for the \( n \)-th term of an AP:
\[
l = a + (n-1)d
\]
where \( l \) is the last term (99 in this case). Plugging in the values:
\[
99 = 3 + (n-1) \cdot 6
\]
Subtracting 3 from both sides:
\[
96 = (n-1) \cdot 6
\]
Dividing both sides by 6:
\[
n-1 = 16
\]
Thus, \( n = 17 \).
### Step 4: Calculate the sum of the series using the sum formula for an AP.
The sum \( S_n \) of the first \( n \) terms of an AP is given by:
\[
S_n = \frac{n}{2} \cdot (2a + (n-1)d)
\]
Substituting \( n = 17 \), \( a = 3 \), and \( d = 6 \):
\[
S_{17} = \frac{17}{2} \cdot (2 \cdot 3 + (17-1) \cdot 6)
\]
Calculating inside the parentheses:
\[
S_{17} = \frac{17}{2} \cdot (6 + 96)
\]
\[
S_{17} = \frac{17}{2} \cdot 102
\]
\[
S_{17} = \frac{17 \cdot 102}{2}
\]
Calculating \( 17 \cdot 102 = 1734 \):
\[
S_{17} = \frac{1734}{2} = 867
\]
### Final Answer:
The sum of all odd numbers between 1 and 100 that are divisible by 3 is **867**.
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