Find Consecutive Numbers by Average Calculator – Natural, Even, Odd

This is a free online calculator for finding consecutive numbers based on a given average. The consecutive numbers can be consecutive integers, consecutive even numbers, or consecutive odd numbers.

Here, the consecutive numbers refer to consecutive integers. Consecutive integers means that these numbers are all integers, they are next to each other, and there are no other integers between them.

Suppose there are N consecutive numbers, the first number is X, and their mean is Y.

Because consecutive numbers are an arithmetic sequence, the tolerance is 1. Therefore, according to the formula for the sum of arithmetic sequences

S_n = a₁ * n + n * (n – 1) * d2

Among them, a₁ represents the first item of the sequence, n represents the number of items, and d is the tolerance.

The sum of these N consecutive numbers is

S_n = X * N + N * (N – 1)2

Their average is

S_nN = X + N – 12

Since the mean of these consecutive numbers is Y, we can get an equation

X + N – 12 = Y

Simplifying the equation, we can get the formula for the first number of consecutive integers.

X + N – 12 = Y

X = Y – N – 12

Where Y represents the average value, and N represents the number of consecutive numbers. X is an integer. If the calculated value of X is a decimal, it means that there are no consecutive integers that satisfy the condition.

For example, the average of 5 consecutive numbers is 64. What are these 5 numbers?

The first number of them is

X = Y – N – 12

X = 64 – 5 – 12

X = 64 – 2

X = 63

So, these 5 consecutive numbers are 63, 64, 65, 66 and 67.

Let’s look at another example with no results.

The average of 4 consecutive numbers is 64. What are these 4 numbers?

Similarly, according to the formula to calculate

X = Y – N – 12

X = 64 – 4 – 12

X = 64 – 1.5

X = 62.5

Here, the value of X is a decimal. Therefore, there are no 4 consecutive integers with an average of 64.

Consecutive odd numbers and consecutive even numbers are both arithmetic sequences with a tolerance of 2.

Again, following the above steps, we can get the formula for its sum.

S_n = X * N + N * (N – 1)

The mean of them is

S_nN = X + (N – 1)

So, the formula for the first number of consecutive even/odd numbers is

Y = X + (N – 1)

X = Y – N + 1

Where Y represents the average value, and N represents the number of consecutive even/odd numbers.

For example, the average of 4 consecutive even numbers is 64. What are these consecutive even numbers?

X = 64 – 4 + 1 = 61

Because X is even, but the result of the calculation is odd. So, there are no 4 consecutive even numbers whose average is 64.

If the average of 4 consecutive even numbers is 65.

X = 65 – 4 + 1 = 62

So, these 4 consecutive even numbers are 62, 64, 66 and 68.

Verify

62 + 64 + 66 + 684 = 2604 = 65

Correct.

The procedure to use the calculator is as follows:

Next, let’s look at some examples.