Question

Find a real number \(c\) and an \(N_{0} \in \mathbb{N}\) such that \(n^{2}+5 n

Short answer

Choose \(c = 2\) and \(N_0 = 6\).

Step-by-step solution

Simplify the Given Inequality

We start with the inequality given in the problem: \( n^2 + 5n < c n^2 \). This can be rewritten by moving all terms to one side of the inequality: \( 5n < (c-1)n^2 \).

Factor and Simplify the Expression

Factor the expression by dividing both sides by \(n\) (this is valid as long as \(n eq 0\), which is true for \(n \in \mathbb{N}\)): \( 5 < (c-1)n \). This further simplifies to \(\frac{5}{n} < c-1\).

Choose Values for \(c\) and \(N_0\)

We start by selecting \(c\) such that \(c \gt 1\). Let's take \(c = 2\). Substitute into the inequality: \(\frac{5}{n} < 2-1\), which simplifies to \(\frac{5}{n} < 1\).

Solve for \(N_0\)

For the inequality \(\frac{5}{n} < 1\) to hold, multiply both sides by \(n\) (valid for \(n > 0\)): \(5 < n\). Hence, \(n \geq 6\) satisfies the inequality. Therefore, we choose \(N_0 = 6\).

Verification

Verify that for \(n \geq 6\), the inequality \( n^2 + 5n < 2n^2 \) holds. For \(n = 6\), we have \(n^2 + 5n = 36 + 30 = 66\) and \(2n^2 = 72\). Thus, \(66 < 72\), confirming our inequality is satisfied.

Key concepts

Algebraic Manipulation

Algebraic manipulation refers to the ability to rearrange and simplify algebraic expressions and equations. It involves using operations such as addition, subtraction, multiplication, and division to transform one expression into another, more useful form. In this exercise, we start with an inequality: \(n^2 + 5n < cn^2\). The goal is to manipulate this expression to find values for \(c\) and \(N_0\) that satisfy the conditions given.

By subtracting \(n^2\) from both sides, the original inequality \(n^2 + 5n < cn^2\) becomes \(5n < (c-1)n^2\). At this point, factoring plays a key role by dividing both sides by \(n\), assuming \(n eq 0\), resulting in \(5 < (c-1)n\). This step showcases the power of algebraic manipulation in helping us better understand and solve inequalities.

Algebraic manipulation is a fundamental skill in mathematics as it allows for complex problems to be broken down into simpler, solvable parts.

Real Numbers

Real numbers encompass all the numbers on the number line, including whole numbers, fractions, decimals, and irrationals. They are integral to many areas of mathematics, including discrete mathematics. In this exercise, the challenge is to find a real number \(c\), where \(c \gt 1\).

Our chosen \(c\) must ensure the inequality \(5n < (c-1)n^2\) is satisfied for all \(n\) in natural numbers, starting from a particular \(N_0\). By selecting \(c = 2\), the inequality becomes \(5n < n^2\), which further simplifies to \(n \gt 5\) after dividing through by \(n\). This demonstrates the flexibility and breadth of real numbers in finding solutions to inequalities.

Real numbers are crucial in defining continuous quantities and can help us understand various mathematical phenomena by allowing us the freedom to choose the most suitable values in different contexts.

Natural Numbers

Natural numbers are the set of positive integers starting from 1, denoted as \(\mathbb{N}\). They are used to count and order, making them foundational in discrete mathematics. For this exercise, we needed to determine a natural number \(N_0\) such that the inequality \(n^2 + 5n < cn^2\) holds for all \(n \geq N_0\).

After transforming the inequality to \(5 < (c-1)n\), and selecting \(c = 2\), it simplifies to \(5 < n\). Thus, \(N_0 = 6\) becomes the smallest natural number for which the inequality holds true. This choice shows how natural numbers help establish bounds and thresholds in mathematical reasoning.

These numbers are often used to define discrete structures and shapes, which are crucial in various applications of discrete mathematics.

Mathematical Reasoning

Mathematical reasoning involves using logical steps to solve problems or prove statements. It comprises making conjectures, forming arguments, and proving interpretations relying on given assumptions and rules. In this problem, reasoning is employed at each step to ensure consistency and correctness of the solution.

From simplifying \(n^2 + 5n < cn^2\) to validating the choice \(c = 2\) and \(N_0 = 6\), every step depends on logical deductions. Mathematical reasoning helps us check our findings, like verifying the inequality for \(n = 6\), ensuring that \(66 < 72\).

This approach builds not only confidence in the solution but cultivates an analytical mindset essential for tackling more complex mathematical challenges.