sum of those integers from 1 to 500 which are multiples of 2 or 5

Sure, let's break this down:

1. First, we need to find the sum of integers from 1 to 500 which are multiples of 2.

2. The multiples of 2 in this range are 2, 4, 6, ..., 500. This is an arithmetic sequence where the first term (a) is 2, the last term (l) is 500 and the common difference (d) is 2. The number of terms (n) can be calculated as ((l - a)/d) + 1 = ((500 - 2)/2) + 1 = 250. The sum (S) of an arithmetic sequence can be calculated as n/2 * (a + l) = 250/2 * (2 + 500) = 125 * 502 = 62750.

3. Next, we need to find the sum of integers from 1 to 500 which are multiples of 5.

4. The multiples of 5 in this range are 5, 10, 15, ..., 500. This is another arithmetic sequence where a is 5, l is 500 and d is 5. The number of terms n can be calculated as ((l - a)/d) + 1 = ((500 - 5)/5) + 1 = 100. The sum S can be calculated as n/2 * (a + l) = 100/2 * (5 + 500) = 50 * 505 = 25250.

5. However, we have counted numbers which are multiples of both 2 and 5 (i.e., multiples of 10) twice. So, we need to subtract the sum of these numbers once.

6. The multiples of 10 in this range are 10, 20, 30, ..., 500. This is another arithmetic sequence where a is 10, l is 500 and d is 10. The number of terms n can be calculated as ((l - a)/d) + 1 = ((500 - 10)/10) + 1 = 50. The sum S can be calculated as n/2 * (a + l) = 50/2 * (10 + 500) = 25 * 510 = 12750.

7. So, the sum of integers from 1 to 500 which are multiples of 2 or 5 is 62750 + 25250 - 12750 = 75250.