Find the sum of all the integers between 100 and 1000 that are divisibleby 9.

Sure, here's how you can find the sum of all the integers between 100 and 1000 that are divisible by 9:

Step 1: Find the first and the last number divisible by 9 in the given range.

The first number greater than 100 that is divisible by 9 is 108 (since 100/9 = 11 remainder 1, so the next multiple of 9 is 11*9 + 9 = 108).

The last number less than 1000 that is divisible by 9 is 999 (since 1000/9 = 111 remainder 1, so the last multiple of 9 is 111*9 = 999).

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

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

In this case, l = 999, a = 108, and d = 9. So, n = (999 - 108)/9 + 1 = 100.

Step 3: Find the sum of the arithmetic sequence.

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

In this case, S = 100/2 * (108 + 999) = 50 * 1107 = 55350.

So, the sum of all the integers between 100 and 1000 that are divisible by 9 is 55350.

Sure, here is the step by step solution:

Step 1: Find the first and the last number divisible by 9 in the given range.

The first number greater than 100 that is divisible by 9 is 108 (since 100 ÷ 9 = 11 remainder 1, so the next multiple of 9 is 11*9 + 9 = 108).

The last number less than 1000 that is divisible by 9 is 999 (since 1000 ÷ 9 = 111 remainder 1, so the last multiple of 9 is 111*9 = 999).

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

The formula to find the number of terms in an arithmetic sequence is: n = (l - a)/d + 1

where: n = number of terms l = last term a = first term d = common difference (in this case, 9)

So, n = (999 - 108)/9 + 1 = 100

Step 3: Find the sum of the arithmetic sequence.

The formula to find the sum of an arithmetic sequence is: S = n/2 * (a + l)

where: S = sum of the sequence n = number of terms a = first term l = last term

So, S = 100/2 * (108 + 999) = 50 * 1107 = 55350

So, the sum of all the integers between 100 and 1000 that are divisible by 9 is 55350.