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: Find the multiples of 4 from 1 to 520 The multiples of 4 are 4, 8, 12, ..., up to 520. This is an arithmetic sequence with a common difference of 4. The first term (a) is 4 and the last term (l) is 520.

The sum (S) of an arithmetic sequence can be found using the formula S = n/2 * (a + l), where n is the number of terms.

To find n, we use the formula n = (l - a)/d + 1, where d is the common difference. Here, n = (520 - 4)/4 + 1 = 130.

So, the sum of all multiples of 4 from 1 to 520 is S = 130/2 * (4 + 520) = 34120.

Step 2: Find the multiples of 5 from 1 to 520 The multiples of 5 are 5, 10, 15, ..., up to 520. This is an arithmetic sequence with a common difference of 5. The first term (a) is 5 and the last term (l) is 520.

To find n, we use the formula n = (l - a)/d + 1, where d is the common difference. Here, n = (520 - 5)/5 + 1 = 104.

So, the sum of all multiples of 5 from 1 to 520 is S = 104/2 * (5 + 520) = 27300.

Step 3: Find the multiples of 20 from 1 to 520 The multiples of 20 are 20, 40, 60, ..., up to 520. This is an arithmetic sequence with a common difference of 20. The first term (a) is 20 and the last term (l) is 520.

To find n, we use the formula n = (l - a)/d + 1, where d is the common difference. Here, n = (520 - 20)/20 + 1 = 26.

So, the sum of all multiples of 20 from 1 to 520 is S = 26/2 * (20 + 520) = 7020.

Step 4: Add the sums from steps 1 and 2, then subtract the sum from step 3 The sum of all integers from 1 to 520 that are multiples of 4 or 5 is 34120 + 27300 - 7020 = 54400.

This is because the multiples of 20 were counted twice (once in the multiples of 4 and once in the multiples of 5), so we subtract them once to correct for this.

Sure, let's break this down:

1. First, we need to find all the multiples of 4 from 1 to 520. The first multiple of 4 is 4 and the last multiple of 4 in this range is 520. We can use the formula for the sum of an arithmetic series to find the sum of these numbers. The formula is:

   Sum = n/2 * (a + l)

   where n is the number of terms, a is the first term, and l is the last term.

   The number of terms (n) can be found by dividing the last term by the common difference (4 in this case), so n = 520/4 = 130.

   So, the sum of multiples of 4 is:

   Sum_4 = 130/2 * (4 + 520) = 34120

2. Next, we need to find all the multiples of 5 from 1 to 520. The first multiple of 5 is 5 and the last multiple of 5 in this range is 520. Using the same formula as above, we find:

   The number of terms (n) is 520/5 = 104.

   So, the sum of multiples of 5 is:

   Sum_5 = 104/2 * (5 + 520) = 27300

3. However, we have counted numbers that are multiples of both 4 and 5 (i.e., multiples of 20) twice, so we need to subtract these out. The first multiple of 20 is 20 and the last multiple of 20 in this range is 520. Using the same formula as above, we find:

   The number of terms (n) is 520/20 = 26.

   So, the sum of multiples of 20 is:

   Sum_20 = 26/2 * (20 + 520) = 7020

4. Therefore, the sum of all integers from 1 to 520 that are multiples of 4 or 5 is:

   Sum = Sum_4 + Sum_5 - Sum_20 = 34120 + 27300 - 7020 = 54400

Sure, let's break this down into steps:

Step 1: Find the sum of all integers from 1 to 520 that are multiples of 4.

The multiples of 4 between 1 and 520 are 4, 8, 12, ..., 520. This is an arithmetic sequence where the first term (a) is 4, the common difference (d) is 4, and we need to find the last term (l) and the number of terms (n).

The last term (l) is 520.

To find the number of terms (n), we use the formula of the nth term of an arithmetic sequence, which is a + (n - 1)d = l. Substituting the known values, we get 4 + (n - 1)4 = 520. Solving for n, we get n = 130.

The sum (S) of an arithmetic sequence is given by the formula S = n/2 * (a + l). Substituting the known values, we get S = 130/2 * (4 + 520) = 34120.

Step 2: Find the sum of all integers from 1 to 520 that are multiples of 5.

The multiples of 5 between 1 and 520 are 5, 10, 15, ..., 520. This is an arithmetic sequence where the first term (a) is 5, the common difference (d) is 5, and we need to find the last term (l) and the number of terms (n).

The last term (l) is 520.

To find the number of terms (n), we use the formula of the nth term of an arithmetic sequence, which is a + (n - 1)d = l. Substituting the known values, we get 5 + (n - 1)5 = 520. Solving for n, we get n = 104.

The sum (S) of an arithmetic sequence is given by the formula S = n/2 * (a + l). Substituting the known values, we get S = 104/2 * (5 + 520) = 27300.

Step 3: Subtract the sum of integers that are multiples of both 4 and 5 (i.e., multiples of 20) because they are counted twice in the above two steps.

The multiples of 20 between 1 and 520 are 20, 40, 60, ..., 520. This is an arithmetic sequence where the first term (a) is 20, the common difference (d) is 20, and we need to find the last term (l) and the number of terms (n).

The last term (l) is 520.

To find the number of terms (n), we use the formula of the nth term of an arithmetic sequence, which is a + (n - 1)d = l. Substituting the known values, we get 20 + (n - 1)20 = 520. Solving for n, we get n = 26.

The sum (S) of an arithmetic sequence is given by the formula S = n/2 * (a + l). Substituting the known values, we get S = 26/2 * (20 + 520) = 7020.

Step 4: The sum of all integers from 1 to 520 that are multiples of 4 or 5 is the sum from step 1 plus the sum from step 2 minus the sum from step 3, which is 34120 + 27300 - 7020 = 54400.

Sure, let's break this down:

1. First, we need to find all the multiples of 4 from 1 to 520. The first multiple of 4 is 4 and the last multiple of 4 in this range is 520. The total number of terms in this arithmetic progression is 520/4 = 130.

   The sum of an arithmetic progression is given by the formula: n/2 * (a + l) where n is the number of terms, a is the first term, and l is the last term.

   So, the sum of multiples of 4 is: 130/2 * (4 + 520) = 34120.

2. Next, we need to find all the multiples of 5 from 1 to 520. The first multiple of 5 is 5 and the last multiple of 5 in this range is 520. The total number of terms in this arithmetic progression is 520/5 = 104.

   Using the same formula, the sum of multiples of 5 is: 104/2 * (5 + 520) = 27300.

3. However, we have counted numbers that are multiples of both 4 and 5 (i.e., multiples of 20) twice, so we need to subtract these out. The first multiple of 20 is 20 and the last multiple of 20 in this range is 520. The total number of terms in this arithmetic progression is 520/20 = 26.

   Using the same formula, the sum of multiples of 20 is: 26/2 * (20 + 520) = 7020.

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

Sure, let's break this down:

1. First, we need to find all the multiples of 4 from 1 to 520. The first multiple of 4 is 4 and the last multiple of 4 in this range is 520. We can use the formula for the sum of an arithmetic series to find the sum of these numbers. The formula is:

   Sum = n/2 * (a + l)

   where n is the number of terms, a is the first term, and l is the last term.

   The number of terms (n) can be found by dividing the last term by the common difference (4 in this case), so n = 520/4 = 130.

   So, the sum of multiples of 4 is: Sum_4 = 130/2 * (4 + 520) = 34120.

2. Next, we need to find all the multiples of 5 from 1 to 520. The first multiple of 5 is 5 and the last multiple of 5 in this range is 520. Using the same formula as above, we find:

   The number of terms (n) is 520/5 = 104.

   So, the sum of multiples of 5 is: Sum_5 = 104/2 * (5 + 520) = 27300.

3. However, we have counted some numbers twice - those that are multiples of both 4 and 5, i.e., multiples of 20. We need to find these and subtract them from our total. The first multiple of 20 is 20 and the last multiple of 20 in this range is 520. Using the same formula as above, we find:

   The number of terms (n) is 520/20 = 26.

   So, the sum of multiples of 20 is: Sum_20 = 26/2 * (20 + 520) = 7020.

4. Finally, we add the sums of multiples of 4 and 5, and subtract the sum of multiples of 20 to get our answer:

   Sum = Sum_4 + Sum_5 - Sum_20 = 34120 + 27300 - 7020 = 54400.

Sure, let's break this down:

1. First, we need to find the sum of all integers from 1 to 520 that are multiples of 4.

2. The smallest multiple of 4 in this range is 4 and the largest is 520. So, we have an arithmetic sequence with a first term (a) of 4, a last term (l) of 520 and a common difference (d) of 4.

3. The number of terms (n) in this sequence can be found using the formula n = ((l - a) / d) + 1. Substituting the given values, we get n = ((520 - 4) / 4) + 1 = 130.

4. The sum (S) of an arithmetic sequence can be found using the formula S = n/2 * (a + l). Substituting the given values, we get S = 130/2 * (4 + 520) = 34120.

5. Next, we need to find the sum of all integers from 1 to 520 that are multiples of 5.

6. The smallest multiple of 5 in this range is 5 and the largest is 520. So, we have an arithmetic sequence with a first term (a) of 5, a last term (l) of 520 and a common difference (d) of 5.

7. The number of terms (n) in this sequence can be found using the formula n = ((l - a) / d) + 1. Substituting the given values, we get n = ((520 - 5) / 5) + 1 = 104.

8. The sum (S) of an arithmetic sequence can be found using the formula S = n/2 * (a + l). Substituting the given values, we get S = 104/2 * (5 + 520) = 27300.

9. However, we have counted numbers that are multiples of both 4 and 5 (i.e., multiples of 20) twice. So, we need to subtract the sum of these numbers from our total.

10. The smallest multiple of 20 in this range is 20 and the largest is 520. So, we have an arithmetic sequence with a first term (a) of 20, a last term (l) of 520 and a common difference (d) of 20.

11. The number of terms (n) in this sequence can be found using the formula n = ((l - a) / d) + 1. Substituting the given values, we get n = ((520 - 20) / 20) + 1 = 26.

12. The sum (S) of an arithmetic sequence can be found using the formula S = n/2 * (a + l). Substituting the given values, we get S = 26/2 * (20 + 520) = 7020.

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

Sure, let's solve this step by step.

1. First, we need to find the sum of all integers from 1 to 520 that are multiples of 4.

   The multiples of 4 between 1 and 520 are 4, 8, 12, ..., 520. This is an arithmetic sequence with a common difference of 4. The first term (a) is 4 and the last term (l) is 520.

   The sum (S) of an arithmetic sequence can be found using the formula S = n/2 * (a + l), where n is the number of terms.

   The number of terms can be found by the formula n = (l - a)/d + 1, where d is the common difference.

   So, n = (520 - 4)/4 + 1 = 130.

   Therefore, the sum of all multiples of 4 from 1 to 520 is S = 130/2 * (4 + 520) = 34120.

2. Next, we need to find the sum of all integers from 1 to 520 that are multiples of 5.

   The multiples of 5 between 1 and 520 are 5, 10, 15, ..., 520. This is an arithmetic sequence with a common difference of 5. The first term (a) is 5 and the last term (l) is 520.

   The number of terms can be found by the formula n = (l - a)/d + 1, where d is the common difference.

   So, n = (520 - 5)/5 + 1 = 104.

   Therefore, the sum of all multiples of 5 from 1 to 520 is S = 104/2 * (5 + 520) = 27300.

3. However, we have counted the numbers that are multiples of both 4 and 5 (i.e., multiples of 20) twice. So, we need to subtract the sum of these numbers once.

   The multiples of 20 between 1 and 520 are 20, 40, 60, ..., 520. This is an arithmetic sequence with a common difference of 20. The first term (a) is 20 and the last term (l) is 520.

   The number of terms can be found by the formula n = (l - a)/d + 1, where d is the common difference.

   So, n = (520 - 20)/20 + 1 = 26.

   Therefore, the sum of all multiples of 20 from 1 to 520 is S = 26/2 * (20 + 520) = 7020.

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

Sure, let's break this down:

1. First, we need to find all the multiples of 4 from 1 to 520. The first multiple of 4 is 4 and the last multiple of 4 in this range is 520. The total number of terms in this sequence is 520/4 = 130.

   We can use the formula for the sum of an arithmetic series to find the sum of these terms: (first term + last term) * number of terms / 2. So, the sum of all multiples of 4 from 1 to 520 is (4 + 520) * 130 / 2 = 34120.

2. Next, we need to find all the multiples of 5 from 1 to 520. The first multiple of 5 is 5 and the last multiple of 5 in this range is 520. The total number of terms in this sequence is 520/5 = 104.

   Using the same formula as above, the sum of all multiples of 5 from 1 to 520 is (5 + 520) * 104 / 2 = 27300.

3. However, we have counted some numbers twice - those that are multiples of both 4 and 5, i.e., multiples of 20. We need to find these and subtract them from our total.

   The first multiple of 20 is 20 and the last multiple of 20 in this range is 520. The total number of terms in this sequence is 520/20 = 26.

   Using the same formula as above, the sum of all multiples of 20 from 1 to 520 is (20 + 520) * 26 / 2 = 7020.

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