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 divisible by 3 within this range are:
- 3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93, 99.
### Step 2: Determine the first term (A), common difference (D), and last term (L).
- The first term (A) is 3.
- The common difference (D) is 6 (since the next odd number divisible by 3 is obtained by adding 6).
- The last term (L) is 99.
### Step 3: Find the number of terms (n) in this sequence.
We can use the formula for the n-th term of an arithmetic progression:
\[ T_n = A + (n - 1) \cdot D \]
Setting \( T_n = 99 \):
\[ 99 = 3 + (n - 1) \cdot 6 \]
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 arithmetic series.
The sum \( S_n \) of the first n terms of an arithmetic series can be calculated using the formula:
\[ S_n = \frac{n}{2} \cdot (A + L) \]
Substituting the values we found:
\[ S_{17} = \frac{17}{2} \cdot (3 + 99) \]
Calculating inside the parentheses:
\[ S_{17} = \frac{17}{2} \cdot 102 \]
Calculating further:
\[ S_{17} = 17 \cdot 51 = 867 \]
### Final Answer:
The sum of all odd numbers between 1 and 100 that are divisible by 3 is **867**.
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