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

Options

A. 83667
B. 90000
C. 83660
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 between 1 and 1000 that are divisible by 3.
The first odd number that is divisible by 3 is 3. The next odd number that is divisible by 3 can be found by adding 6 (since we are looking for odd numbers, and the next odd number after an odd number divisible by 3 is 6 units away).

So, 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 odd number less than or equal to 1000 that is divisible by 3, we can check:
- The largest number divisible by 3 less than or equal to 1000 is 999.
- Since 999 is odd, it is the last term in our sequence.

### Step 3: Determine the number of terms in the sequence.
The sequence of odd numbers divisible by 3 can be expressed as:
3, 9, 15, 21, ..., 999.

This is an arithmetic progression (AP) where:
- First term (a) = 3
- Common difference (d) = 6
- Last term (l) = 999

To find the number of terms (n), we can use the formula for the nth term of an AP:
\[ a_n = a + (n - 1) \cdot d \]
Setting \( a_n = 999 \):
\[ 999 = 3 + (n - 1) \cdot 6 \]
\[ 999 - 3 = (n - 1) \cdot 6 \]
\[ 996 = (n - 1) \cdot 6 \]
\[ n - 1 = \frac{996}{6} \]
\[ n - 1 = 166 \]
\[ n = 167 \]

### Step 4: Calculate the sum of the series.
The sum \( S_n \) of the first n terms of an AP is given by:
\[ S_n = \frac{n}{2} \cdot (a + l) \]
Substituting the values we have:
\[ S_{167} = \frac{167}{2} \cdot (3 + 999) \]
\[ S_{167} = \frac{167}{2} \cdot 1002 \]
\[ S_{167} = 167 \cdot 501 \]
\[ S_{167} = 83667 \]

### Final Answer:
The sum of all odd numbers between 1 and 1000 which are divisible by 3 is **83667**.

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