In mathematics, you often come across problem statements wherein you are required to find two or more consecutive integers if their sum or difference is given to you. A number of times the clause of these consecutive numbers being odd or even is also added. Before simplifying the problem statement and forming its equation, you are supposed to use a variable for one integer and then represent it in the form of its consecutive integers. Let me explain this with the help of an example.

Say you have to find two consecutive integers whose sum is 89. How do you go about this problem? You first take a variable, say x, the value of which is unknown to you. Next, you are supposed to take another number. Since the problem requires you to use two consecutive integers, the integer next to x will be (x + 1). Now as per the problem, the sum of x and (x+1) is 89. We represent this in the form of an equation: x + (x + 1) = 89. Solving this equation we get x as 44 and the next integer (x + 1) as 45, the sum of which is 89.

Similarly, knowing the consecutive integer formula finds application in a number of mathematical problems. In this section, we will go through some of these formulas.

Consecutive Integer Formula

If n is an integer, (n + 1) and (n + 2) will be the next two consecutive integers. For example, let n be 1. We find its consecutive integers as (1 + 1) and (1 + 2), or 2 and 3.

Hence, the formula:

n, n+1, n+2, n+3,…

**Even Consecutive Integer Formula** In mathematics, we represent an even integer as 2n. If 2n is an even integer, (2n + 2) and (2n + 4) will be the next two even consecutive integers. For example, let 2n be 4, which is an even integer. We find its consecutive integers as (4 + 2) and (4 + 4), or 6 and 8.

Hence, the formula:

2n, 2n+2, 2n+4, 2n+6,…

Note that the difference between two even consecutive integers here is 2, otherwise, we would end up with an integer which is consecutive but not even.

**Odd Consecutive Integer Formula** In mathematics, we represent an odd integer as 2n + 1. If 2n + 1 is an odd integer, (2n + 3) and (2n + 5) will be the next two odd consecutive integers. For example, let 2n + 1 be 7, which is an odd integer. We find its consecutive integers as (7 + 2) and (7 + 4), or 9 and 11.

Hence, the formula:

2n+1, 2n+3, 2n+5, 2n+7,…

Note that the difference between two odd consecutive integers here is 2, otherwise, we would end up with an integer which is consecutive but not odd.

Solved Examples

Question: Find three consecutive integers of 76.

Solution:

Let 76 be n. So the next three consecutive integers will be n + 1, n + 2 and n + 3.

76 + 1, 76 + 2 and 76 + 3 or 77, 78 and 79

Therefore, we have 76, 77, 78, 79

Question: Find three consecutive even integers of -8.

Solution:

Let -8 be 2n. So the next three consecutive integers will be 2n + 2, 2n + 4 and 2n + 6.

-8 + 2, -8 + 4 and -8 + 6 or -6, -4 and -2

Therefore, we have -8, -6, -4, -2