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 previous multiple of 7 is 142*7 = 994).

Step 2: Find the number of terms in the sequence.

The number of terms in the sequence can be found using the formula for the nth term of an arithmetic sequence, which is a + (n - 1)d, where a is the first term, d is the common difference, and n is the number of terms. Here, a = 105, d = 7, and the nth term = 994.

So, 994 = 105 + (n - 1)7. Solving for n gives n = 128.

Step 3: Find the sum of the 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, n = 128, a = 105, and l = 994.

So, S = 128/2 * (105 + 994) = 64 * 1099 = 70336.

So, the sum of all integers between 100 and 1000 which are divisible by 7 is 70336.