Problem 17
Question
Write an algebraic formula for the given quantity.. The product \(P\) of two consecutive integers, the first integer being \(n\)
Step-by-Step Solution
Verified Answer
The formula is \( P = n(n+1) \).
1Step 1: Identify the First Integer
The problem states that the first integer is given as \( n \). So, the first integer is \( n \).
2Step 2: Determine the Second Integer
Since the problem involves two consecutive integers, the second integer is the next integer after \( n \). Therefore, the second integer can be expressed as \( n+1 \).
3Step 3: Express the Product of the Integers
The product \( P \) of the two integers is the result of multiplying them together. This can be written as \( P = n \times (n+1) \).
Key Concepts
Consecutive integersProduct of integersAlgebraic formula
Consecutive integers
Consecutive integers are numbers that follow each other in order and have a difference of 1 between them. They are like steps on a staircase, where each new step is one unit higher than the last. When we talk about consecutive integers, we use terms like "the first integer," "the next integer," and so on.
For example, if we start with the integer 5, the next consecutive integer after 5 is 6. In a sequence, we could have integers like 3, 4, 5, and so forth.
In mathematical expressions, if we let the first integer be represented by a variable like "\(n\)", the next consecutive integer would naturally be "\(n+1\)". This is very helpful in algebra because it allows us to generalize and solve problems involving a series of consecutive numbers easily.
For example, if we start with the integer 5, the next consecutive integer after 5 is 6. In a sequence, we could have integers like 3, 4, 5, and so forth.
In mathematical expressions, if we let the first integer be represented by a variable like "\(n\)", the next consecutive integer would naturally be "\(n+1\)". This is very helpful in algebra because it allows us to generalize and solve problems involving a series of consecutive numbers easily.
Product of integers
The concept of the
For instance, if you multiply 3 and 4, the product is 12. The numbers being multiplied, in this case, 3 and 4, are called "factors."
In the context of the solution provided, the product of consecutive integers is particularly focused on. When you have two integers like \(n\) and \(n+1\), their product is the multiplication of these two numbers, represented algebraically as \(P = n \times (n+1)\). This expression is powerful because it shows the relationship between these consecutive numbers instantly via a simple multiplication.
- Product
- Factors
- Multiplication
For instance, if you multiply 3 and 4, the product is 12. The numbers being multiplied, in this case, 3 and 4, are called "factors."
In the context of the solution provided, the product of consecutive integers is particularly focused on. When you have two integers like \(n\) and \(n+1\), their product is the multiplication of these two numbers, represented algebraically as \(P = n \times (n+1)\). This expression is powerful because it shows the relationship between these consecutive numbers instantly via a simple multiplication.
Algebraic formula
An algebraic formula is an equation that shows a relationship between different variables. These formulas are like recipes in cooking. They provide a consistent method to calculate or solve for different elements.
In the exercise, we're looking at the formula for the product of two consecutive integers. The formula is expressed as \(P = n \times (n+1)\), where \(P\) is the product, \(n\) is the first integer, and \(n+1\) is the next consecutive integer.
This formula is useful in various mathematical problems as it allows you to find the product by simply plugging in the value of \(n\). It embodies a clear mathematical relationship and simplifies complex arithmetic into concise algebraic expressions. Remember, algebraic formulas can save you lots of time and effort, especially with more complicated numbers or longer series of integers.
In the exercise, we're looking at the formula for the product of two consecutive integers. The formula is expressed as \(P = n \times (n+1)\), where \(P\) is the product, \(n\) is the first integer, and \(n+1\) is the next consecutive integer.
This formula is useful in various mathematical problems as it allows you to find the product by simply plugging in the value of \(n\). It embodies a clear mathematical relationship and simplifies complex arithmetic into concise algebraic expressions. Remember, algebraic formulas can save you lots of time and effort, especially with more complicated numbers or longer series of integers.
Other exercises in this chapter
Problem 17
17–24 ? Use a Factoring Formula to factor the expression. $$ 9 a^{2}-16 $$
View solution Problem 17
Evaluate each expression. (a) \(100^{-1.5}\) (b) \(4^{2 / 3} \cdot 6^{2 / 3} \cdot 9^{2 / 3}\) (c) \(0.001^{-2 / 3}\)
View solution Problem 17
\(15-20\) : Use properties of real numbers to write the expression without parentheses. $$ 4(2 m) $$
View solution Problem 18
Evaluate each expression. $$ \left(\frac{1}{2}\right)^{4} \cdot\left(\frac{5}{2}\right)^{2} $$
View solution