The sum of all integers from 1 to 520 that are multiples of 4 or 5

Sure, let's break this down into steps:

Step 1: Identify the multiples of 4 and 5 within the range 1 to 520.

Multiples of 4: The smallest multiple of 4 in this range is 4 and the largest is 520. The multiples of 4 form an arithmetic sequence with a common difference of 4.

Multiples of 5: The smallest multiple of 5 in this range is 5 and the largest is 520. The multiples of 5 form an arithmetic sequence with a common difference of 5.

Step 2: Calculate the sum of these multiples.

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

For multiples of 4: n = 520/4 = 130, a = 4, l = 520. So, S4 = 130/2 * (4 + 520) = 34120.

For multiples of 5: n = 520/5 = 104, a = 5, l = 520. So, S5 = 104/2 * (5 + 520) = 27300.

Step 3: However, multiples of 20 (which are common to both 4 and 5) have been counted twice, so we need to subtract these from the total.

The smallest multiple of 20 in this range is 20 and the largest is 520. For multiples of 20: n = 520/20 = 26, a = 20, l = 520. So, S20 = 26/2 * (20 + 520) = 7020.

Step 4: The final answer is the sum of the multiples of 4 and 5, minus the sum of the multiples of 20 (to correct for double-counting).

So, the sum of all integers from 1 to 520 that are multiples of 4 or 5 is S4 + S5 - S20 = 34120 + 27300 - 7020 = 54400.