Problem 4
Question
In Exercises \(1-4,\) a statement \(S_{n}\) about the positive integers is given. Write statements \(S_{1}, S_{2},\) and \(S_{3},\) and show that each of these statements is true. \(S_{n}: 3\) is a factor of \(n^{3}-n\).
Step-by-Step Solution
Verified Answer
The three statements \(S_1, S_2,\) and \(S_3\) obtained by substituting \(n=1,2,3\) respectively in the expression \(n^3-n\) are all true as in each case the result can be evenly divided by 3.
1Step 1: Write the Statement \(S_{1}\)
Substitute \(n=1\) in \(S_{n}\) to get \(S_{1}= 1^3 - 1 = 0\). Thus, \(S_{1}\) states that '3 is a factor of 0'. This is true as 0 divided by any non-zero number results in 0.
2Step 2: Write the Statement \(S_{2}\)
Substitute \(n=2\) in \(S_{n}\) to get \(S_{2}= 2^3 - 2 = 6\). Thus, \(S_{2}\) states that '3 is a factor of 6'. This can be verified as 6 divided by 3 gives 2 which is an integer. Hence, this statement is also true.
3Step 3: Write the Statement \(S_{3}\)
Substitute \(n=3\) in \(S_{n}\) to get \(S_{3}= 3^3 - 3 = 24\). Thus, \(S_{3}\) states that '3 is a factor of 24'. This statement is true as 24 divided by 3 equals to 8 which is an integer. Thus, all the three statements \(S_{1}, S_{2},\) and \(S_{3}\) are true.
Key Concepts
FactorizationAlgebraic ExpressionsPositive Integers
Factorization
Factorization is a process used in mathematics to break down expressions into products of simpler expressions or numbers, called factors. When we say that a number or expression can be factored, we mean that it can be divided exactly by one or more numbers. For example, the expression \( n^3 - n \) can be factored because it can be expressed in terms of its factors.
To factorize the expression \( n^3 - n \), we start by recognizing a common factor in each term, which is \( n \). We can then factor \( n \) out of the expression:
To factorize the expression \( n^3 - n \), we start by recognizing a common factor in each term, which is \( n \). We can then factor \( n \) out of the expression:
- \( n^3 - n = n(n^2 - 1) \)
- \( n^2 - 1 = (n - 1)(n + 1) \)
- \( n(n - 1)(n + 1) \)
Algebraic Expressions
Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators. In the context of this exercise, the algebraic expression is \( n^3 - n \). This expression represents the result of cubing a number \( n \) and then subtracting \( n \) itself.
Understanding how to manipulate and simplify algebraic expressions is a key mathematical skill. It involves using rules such as the distributive property, combining like terms, and sometimes performing operations like factoring.
In our case, rewriting \( n^3 - n \) as a product, \( n(n^2 - 1) \), simplifies the analysis of divisibility. By recognizing patterns such as the difference of squares \( n^2 - 1 = (n - 1)(n + 1) \), we gain more insight into the expression's structure and properties, which help verify statements involving numbers and divisibility.
Understanding how to manipulate and simplify algebraic expressions is a key mathematical skill. It involves using rules such as the distributive property, combining like terms, and sometimes performing operations like factoring.
In our case, rewriting \( n^3 - n \) as a product, \( n(n^2 - 1) \), simplifies the analysis of divisibility. By recognizing patterns such as the difference of squares \( n^2 - 1 = (n - 1)(n + 1) \), we gain more insight into the expression's structure and properties, which help verify statements involving numbers and divisibility.
Positive Integers
Positive integers are all the whole numbers greater than zero, such as 1, 2, 3, and so on. The exercise focuses on positive integers because these numbers exhibit specific properties that are important in problems involving divisibility and factorization.
In the statement \( S_{n} : 3 \) is a factor of \( n^3 - n \), \( n \) is any positive integer. The significance of positive integers in this context lies in their role in number theory, where they help us understand the divisibility and factors of different expressions.
When substituting specific positive integers into an expression like \( n^3 - n \), it becomes easier to see patterns and verify general statements. In this exercise, substituting the first few positive integers shows that the resulting numbers are divisible by 3. The practice reinforces understanding of algebraic properties and prepares students for more complex mathematical concepts.
In the statement \( S_{n} : 3 \) is a factor of \( n^3 - n \), \( n \) is any positive integer. The significance of positive integers in this context lies in their role in number theory, where they help us understand the divisibility and factors of different expressions.
When substituting specific positive integers into an expression like \( n^3 - n \), it becomes easier to see patterns and verify general statements. In this exercise, substituting the first few positive integers shows that the resulting numbers are divisible by 3. The practice reinforces understanding of algebraic properties and prepares students for more complex mathematical concepts.
- For \( n=1 \), the result is 0, which is divisible by 3.
- For \( n=2 \), the result is 6, divisible by 3.
- For \( n=3 \), the result is 24, also divisible by 3.
Other exercises in this chapter
Problem 4
Use the formula for \(_{n} P_{t}\) to evaluate each expression. $$ _{10} P_{4} $$
View solution Problem 4
In Exercises 1-8, evaluate the given binomial coefficient. $$\left(\begin{array}{l}11 \\ 1\end{array}\right)$$
View solution Problem 4
Write the first five terms of each geometric sequence. $$ a_{1}=24, \quad r=\frac{1}{3} $$
View solution Problem 4
In Exercises \(1-14\), write the first six terms of cach arithmetic sequence $$a_{1}=-8, d=5$$
View solution