Consecutive integers are two or more integers in sequence, each of which is one more than the integer that precedes it.
Consecutive integers may be represented by n, n + 1, n + 2, ..., where n is an integer.
For example, −3, −2, −1, 0, 1, 2, 3, 4 are consecutive integers.
Consecutive even integers may be represented by 2n, 2n + 2, 2n + 4, ..., where n is an integer.
For example, 0, 2, 4, 6, 8, 10, 12 are consecutive even integers.
Consecutive odd integers may be represented by 2n + 1, 2n + 3, 2n + 5, ..., where n is an integer.
For example, 5, 7, 9, 11, 13 are consecutive odd integers.
Example
The sum of three consecutive integers is less than 87. What is the maximum possible value of the smallest of the three integers? Solution.
Let the consecutive integers be n, n + 1, and n + 2.
Then, their sum is n + (n + 1) + (n + 2) = 3n + 3.
So, 3n + 3 < 87 ⇒ 3n < 84 ⇒ n < 28.
Thus, the maximum possible value of n is 27.
An alternative method to avoid the algebra and inequalities is given below.
One-third of 87 is 29. So, let's guess the three consecutive integers to be 28, 29, 30. Their sum is 87.
The sum must be less than 87. Therefore, the integers must be 27, 28, 29.
