Find the sum of first all integers between 100 and 1000 which are divisible by 7.

Sure, here are the steps to find the sum of all integers between 100 and 1000 which are divisible by 7:

Step 1: Find the first and the last number divisible by 7 in the given range.

The first number greater than 100 that is divisible by 7 is 105 (since 100 ÷ 7 = 14 remainder 2, so the next multiple of 7 is 14*7 + 7 = 105).

The last number less than 1000 that is divisible by 7 is 994 (since 1000 ÷ 7 = 142 remainder 6, so the last multiple of 7 is 142*7 = 994).

Step 2: Find the total number of terms.

The total number of terms in an arithmetic sequence can be found using the formula: n = (l - a)/d + 1, where l is the last term, a is the first term, and d is the common difference.

Here, l = 994, a = 105, and d = 7.

So, n = (994 - 105)/7 + 1 = 127.

Step 3: Find the sum of the arithmetic sequence.

The sum of an arithmetic sequence can be found using the formula: S = n/2 * (a + l), where S is the sum, n is the number of terms, a is the first term, and l is the last term.

Here, S = 127/2 * (105 + 994) = 69985.

So, the sum of all integers between 100 and 1000 which are divisible by 7 is 69985.