# Find Six Consecutive Integers Whose Sum is 519 by 3 Ways

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The sum of six consecutive integers is 519, what the integers? I can easily tell you that the answer are 84, 85, 86, 87, 88 and 89. You must be interested in how to find these 6 consecutive integers whose sum is 519. There are three methods here, let us introduce them one by one below.

## 1. Find six consecutive integers whose sum is 519 by hypothetical method

Assuming that N is used to represent the first integer, so the second integer is N + 1, the third integer is N + 2, the 4th integer is N + 3, the 5th integer is N + 4, the 6th integer is N + 5.

Therefore, the sum of 6 consecutive integers is 519, which can be expressed by the equation

N + (N + 1) + (N + 2) + (N + 3) + (N + 4) + (N + 5) = 519

Solve this equation

N + (N + 1) + (N + 2) + (N + 3) + (N + 4) + (N + 5) = 519

N + N + 1 + N + 2 + N + 3 + N + 4 + N + 5 = 519

6 * N + 1 + 2 + 3 + 4 + 5= 519

6 * N + 15 = 519

6 * N = 519 – 15

6 * N = 504

N = 504 / 6

N = 84

Now, we can get that 84 is the first integer of 6 consecutive integers whose sum is 519. So, the second integer is 85, the third integer is 86, the 4th integer is 87, the 5th integer is 88, the 6th integer is 89.

The answer came out, the sum of 6 consecutive integers is 519, these three integers are 84, 85, 86, 87, 88 and 89.

Verify: 84 + 85 + 86 + 87 + 88 + 89 = 519. Correct!

This is the most common method, let’s look at the second method, which is my favorite method.

## 2. Find six consecutive integers whose sum is 519 by formula method

According to the consecutive integers calculator based on sum, we can know that the sum of M consecutive integers is S, the first integer formula is

First(n) = S / M – (M – 1) / 2

M represents the number of consecutive integers.

S stands for sum.

Back to the problem we want to solve: find six consecutive integers whose sum is 519. In here, M = 6, S = 519. Replace them in the formula to calculate the first integer

First(n) = 519 / 6 – (6 – 1) / 2 = 86.5 – (6 – 1) / 2 = 86.5 – 5 / 2 = 86.5 – 2.5 = 84

The first integer is 84, so, it can be easily calculated that the second integer is 85, the third integer is 86, the 4th integer is 87, the 5th integer is 88, the 6th integer is 89.

Thus, the answer are also 84, 85, 86, 87, 88 and 89. Same as the first method. Then look at the third method, which is the simplest one.

## 3. Find six consecutive integers whose sum is 519 by average method

The principle is very simple, because we want to calculate the consecutive integers, so after calculating the average, we can find the integers near the average.

The sum of 6 consecutive integers is 519, so, the average of these 6 consecutive integers is 519 / 6 = 86.5. The integers around 86.5 are 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94. Now the problem is simple, find 6 consecutive integers from 81 to 94 and their average is 86.5. The answer are 84, 85, 86, 87, 88 and 89.

So the sum of 6 consecutive integers is 519, these integers are 84, 85, 86, 87, 88, 89. The results are consistent with the above two methods. Is it very simple?

## Problems can be sloved by this answer

Now, we have found out these 6 consecutive integers whose sum is 519, some problems can be easily solved.

• The sum of six consecutive integers is 519, the smallest one is 84.
• The sum of six consecutive integers is 519, the greatest integer is 89.
• The sum of six consecutive integers is 519, the average of these integers is 86.5.
• The sum of six consecutive integers is 519, the product of them is 84 * 85 * 86 * 87 * 88 * 89 = 1785203648.
• The sum of six consecutive integers is 519, the sum of their squares is 842 + 852 + 862 + 872 + 882 + 892 = 44911.

## Summarize

On this page, In addition to introducing three methods how to find six consecutive integers whose sum is 519, it also provides a calculator that calculates six consecutive integers based on the sum. If you encounter a similar problem next time, you can directly use this calculator to calculate the answer, which is very convenient.

Of course, if the problem you encounter is more complicated, such as: the number of consecutive integers is not 6, or you need to calculate consecutive odd integers or even integers. You can use our other more advanced sum-based consecutive integers calculator, where you can specify the number of consecutive integers and select consecutive integers type: natural integers, odd integers or even integers. I believe it can help you.

Well, that’s it, the above are three solutions to find six consecutive integers whose sum is 519. It’s not very complicated, but I personally like the second method best, because it can accurately calculate the first integer and the calculation process is very simple. So how about you? Please leave a message and tell me which method you like?