Question

In Exercises 1–10, write an algebraic expression for the verbal expression. The sum of two consecutive natural numbers

Short answer

The algebraic expression for the sum of two consecutive natural numbers is \(2n + 1\).

Step-by-step solution

Understand the Terms

First, you should get familiar with the meanings of important terms in the exercise. 'Natural numbers' are counting numbers (e.g., 1,2,3,...) and 'consecutive' means 'in a row' or 'following each other without gaps'.

Represent the Natural Number

Secondly, pick a variable to represent the first natural number. For instance, it can be denoted as \(n\).

Express the Consecutive Number

The exercise talks about 'consecutive' numbers, which means the next number is immediately after the previous number. So, if the first natural number is \(n\), the next consecutive natural number would be \(n+1\).

Write the Algebraic Expression

The exercise asks for the 'sum' of two consecutive natural numbers. In algebra, the 'sum' means 'addition'. So the algebraic expression for the sum of two consecutive natural numbers is \(n + (n + 1)\). By following the rule of the mathematical operation, it becomes \(2n + 1\).

Key concepts

Natural Numbers

Natural numbers are the basic building blocks of all other numbers in mathematics. They are the numbers we use for counting, such as 1, 2, 3, and so on. These numbers do not include zero, negative numbers, or fractions—only the numbers that we naturally start counting with from one upwards.

In this context, it is important to recognize that every natural number is positive. This distinction helps ensure that when working with natural numbers, calculations stay simple and intuitive. For instance, if we say "natural number," the smallest is always 1, followed by 2, 3, and all larger whole numbers.

When solving algebraic problems like the one in the exercise, understanding what natural numbers are is crucial. It ensures clarity when setting up expressions and ensures that calculations are accurate and relevant.

Consecutive Numbers

Consecutive numbers are numbers that come one after the other in the number sequence. For natural numbers, this means numbers like 1, 2, 3, where each number is one unit larger than the number before it.

When we talk about algebraic expressions involving consecutive numbers, we use the concept of consecutive numbers quite frequently. For example, if you have a number represented by a variable such as \( n \), the next consecutive number will be \( n + 1 \). This is because you are simply adding one to get to the next number in the sequence.

Understanding consecutive numbers helps in creating and solving expressions. It allows for the accurate representation of a sequential relationship between numbers, a key aspect when calculating sums or differences in algebra problems.

Addition in Algebra

Addition in algebra is similar to addition in arithmetic but includes variables, which makes it more flexible and powerful. Instead of just adding constant numbers, algebra allows us to add expressions that include unknown values.

In the provided exercise, you are required to find the sum of two consecutive natural numbers. This is represented in an algebraic form as \( n + (n + 1) \). Breaking this down, \( n \) represents the first number, while \( n + 1 \) represents the next consecutive number.

When we perform the addition, the algebraic expression becomes \( 2n + 1 \). This simply means we added \( n + n + 1 \). By combining like terms (the \( n \) terms), we simplified our expression to its simplest form. This showcases how adding in algebra not only deals with specific numbers but with expressions that might represent a wide range of values.