The sum of all odd numbers between 1 and 1000 which are divisible by 3, is

Options

A. 90000
B. 83667
C. 93660
D. None of these

Answer

To find the sum of all odd numbers between 1 and 1000 that are divisible by 3, we can follow these steps:

### Step 1: Identify the odd numbers divisible by 3
The first odd number greater than 1 that is divisible by 3 is 3. The subsequent odd numbers divisible by 3 can be found by adding 6 (the common difference between consecutive odd multiples of 3).

The sequence of odd numbers divisible by 3 is:
3, 9, 15, 21, ..., up to the largest odd number less than or equal to 1000 that is divisible by 3.

### Step 2: Find the last term in the sequence
To find the last term, we need to find the largest odd number less than or equal to 1000 that is divisible by 3. The largest odd number less than or equal to 1000 is 999.

Now, we check if 999 is divisible by 3:
999 ÷ 3 = 333 (which is an integer)

Thus, the last term \( a_n \) of our sequence is 999.

### Step 3: Identify the first term and common difference
The first term \( a \) is 3, and the common difference \( d \) is 6.

### Step 4: Use the formula for the nth term of an arithmetic progression
The nth term of an arithmetic progression can be calculated using the formula:
\[ a_n = a + (n - 1)d \]

Substituting the known values:
\[ 999 = 3 + (n - 1) \cdot 6 \]

### Step 5: Solve for n
Rearranging the equation:
\[ 999 - 3 = (n - 1) \cdot 6 \]
\[ 996 = (n - 1) \cdot 6 \]
\[ n - 1 = \frac{996}{6} \]
\[ n - 1 = 166 \]
\[ n = 167 \]

### Step 6: Calculate the sum of the arithmetic series
The sum \( S_n \) of the first n terms of an arithmetic series is given by:
\[ S_n = \frac{n}{2} \cdot (a + a_n) \]

Substituting the values we found:
\[ S_{167} = \frac{167}{2} \cdot (3 + 999) \]
\[ S_{167} = \frac{167}{2} \cdot 1002 \]
\[ S_{167} = 167 \cdot 501 \]

### Step 7: Perform the multiplication
Calculating \( 167 \cdot 501 \):
\[ 167 \cdot 501 = 83,667 \]

### Conclusion
Thus, the sum of all odd numbers between 1 and 1000 that are divisible by 3 is **83,667**.

---