Sum of Natural Numbers Divisible by 13 between 500 and 1000

To find the sum of natural numbers between 500 and 1000 which are divisible by 13, we need to follow these steps:

Step 1: Identify the first and last terms in the series
The first term in the series is the smallest natural number between 500 and 1000 which is divisible by 13. To find this, we can use the formula:

First term = Smallest multiple of 13 greater than or equal to 500

The smallest multiple of 13 greater than or equal to 500 is 507. Therefore, the first term in the series is 507.

The last term in the series is the largest natural number between 500 and 1000 which is divisible by 13. To find this, we can use the formula:

Last term = Largest multiple of 13 less than or equal to 1000

The largest multiple of 13 less than or equal to 1000 is 988. Therefore, the last term in the series is 988.

Step 2: Find the number of terms in the series
To find the number of terms in the series, we need to count the number of multiples of 13 between 507 and 988. We can do this by using the formula:

Number of terms = (Last term - First term) / Common difference + 1

In this case, the common difference is also 13, since we are adding 13 to each term to get the next term. Substituting the values, we get:

Number of terms = (988 - 507) / 13 + 1
Number of terms = 37

Therefore, there are 37 natural numbers between 500 and 1000 which are divisible by 13.

Step 3: Find the sum of the terms
To find the sum of the terms, we can use the formula for the sum of an arithmetic series:

Sum = (Number of terms / 2) x (First term + Last term)

Substituting the values, we get:

Sum = (37 / 2) x (507 + 988)
Sum = 18.5 x 1495
Sum = 27557.5

Since we are dealing with natural numbers, we need to round the answer down to the nearest integer. Therefore, the sum of all natural numbers between 500 and 1000 which are divisible by 13 is:

Sum = 28405 (rounded down from 27557.5)

Therefore, option A is the correct answer.