In the realm of mathematics, you frequently encounter problem statements that task you with determining two or more
Consecutive Integers
when their sum or difference is provided. Often, these problem statements specify whether these consecutive numbers should be odd or even. To tackle these problems, you typically introduce a variable to represent one of the integers and then express the others as consecutive integers. Let me illustrate this with an example.
Suppose you need to identify two consecutive integers whose sum equals 89. How do you approach this problem? Initially, you assign a variable, say 'x,' to one of the unknown integers. Subsequently, since the problem calls for two consecutive integers, the integer immediately following 'x' would be represented as (x + 1). In accordance with the problem, the sum of 'x' and (x + 1) should equal 89. This is expressed in the form of an equation: x + (x + 1) = 89. Solving this equation yields 'x' as 44 and the subsequent integer (x + 1) as 45, and together they sum up to 89.
Similarly, the formula for consecutive integers finds application in various mathematical problems. In this section, we will explore some of these formulas.
What are Consecutive Integers?
Whenever we need to enumerate or arrange items in a specific order, consecutive integers come into play. In essence, consecutive integers are whole numbers that follow one another in a sequence, with a consistent fixed difference between them. For instance, when examining the sequence of natural numbers, such as 1, 2, 3, 4, 5, and 6, we observe that each integer is precisely one unit apart from the next. Likewise, we can construct lists of consecutive even integers, consecutive odd integers, and various other arrangements. It's essential to note that the difference between consecutive integers remains constant, and as they are integers, they can take on positive, negative, or zero values, but they do not encompass fractions or decimals.
Consecutive Integer Formula
When dealing with integers, the consecutive integers following a given integer 'n' are typically expressed as (n + 1) and (n + 2). For instance, if we take 'n' to be 1, the consecutive integers are calculated as (1 + 1) and (1 + 2), resulting in 2 and 3.
Therefore, the general formula for representing consecutive integers is:
n, n + 1, n + 2, n + 3, ...
Even Consecutive Integer Formula
In the realm of mathematics, we represent an even integer as 2n. When we have 2n as an even integer, the next two consecutive even integers can be expressed as (2n + 2) and (2n + 4). For instance, if we take 2n to be 4, which is indeed an even integer, we can calculate its consecutive even integers as (4 + 2) and (4 + 4), resulting in 6 and 8.
Thus, the formula for representing consecutive even integers can be written as:
2n, 2n + 2, 2n + 4, 2n + 6, ...
It's important to note that the difference between two even consecutive integers in this sequence is always 2, ensuring that they remain even integers throughout.
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Odd Consecutive Integer Formula
In the realm of mathematics, an odd integer is typically represented as 2n + 1. When we start with 2n + 1 as an odd integer, the next two consecutive odd integers can be expressed as (2n + 3) and (2n + 5). For instance, if we consider 2n + 1 to be 7, which is indeed an odd integer, we can calculate its consecutive odd integers as (7 + 2) and (7 + 4), resulting in 9 and 11.
As a result, the formula for representing consecutive odd integers can be written as:
2n + 1, 2n + 3, 2n + 5, 2n + 7, ...
It's important to note that the difference between two consecutive odd integers in this sequence is always 2, ensuring that they remain odd integers throughout.
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Solved Examples
Question: Determine three consecutive integers that sum up to 76.
Solution:
Let's use 'n' to represent the first integer. Therefore, the next three consecutive integers would be n + 1, n + 2, and n + 3.
Now, we want their sum to equal 76:
n + (n + 1) + (n + 2) + (n + 3) = 76
Combine like terms:
4n + 6 = 76
Subtract 6 from both sides:
4n = 70
Divide by 4:
n = 70 / 4
n = 17.5
Since 'n' must be an integer, these consecutive integers won't work to sum up to 76.
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Question: Find three consecutive even integers that start from -8.
Solution:
Let's represent -8 as 2n since it's an even integer. The next three consecutive even integers would be 2n + 2, 2n + 4, and 2n + 6.
So, we have:
2n + 2 = -8
2n + 4 = -6
2n + 6 = -4
Now, let's find the value of 'n':
2n + 2 = -8
Subtract 2 from both sides:
2n = -8 - 2
2n = -10
Divide by 2:
n = -10 / 2
n = -5
Now, we can find the three consecutive even integers:
2n + 2 = 2(-5) + 2 = -10 + 2 = -8
2n + 4 = 2(-5) + 4 = -10 + 4 = -6
2n + 6 = 2(-5) + 6 = -10 + 6 = -4
Therefore, we have -8, -6, and -4 as the three consecutive even integers starting from -8.
