Find Sum Of Multiples

We are asked to find all the two-digit multiples of $9$ . We can do this by finding the highest and lowest multiples of $9$ . The highest multiple is $9*11=99$ , and the lowest is $9*2=18$ . Using this, we can create a list of numbers - $18, 27, 36, 45, 54, 63, 72, 81, 90, 99$ . We can simplify this list by dividing each number by $9$ , giving us $2, 3, 4, 5, 6, 7, 8, 9, 10, 11$ . Now, all we need to do is add each of the numbers - $2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11=65$ . Finally, we multiply $65 * 9 = \boxed{585}$ .

However...

There is a much simpler way to solve this problem. We have already found out that we need to sum all the multiples of $9$ from $18$ to $99$ , inclusive. This can be represented by - $9( \sum_{i=2}^{11} i )$ When we simplify the expression above, we get $65 * 9 = \boxed{585}$ , which is the same answer as the first solution.