Question

Find the smallest \(n \in \mathbb{N}\) such that \(2 n^{2}+3 n+1

Short answer

The smallest natural number \( n \) is 4.

Step-by-step solution

Understanding the Inequality

We need to find the smallest natural number \( n \) such that the inequality \( 2n^2 + 3n + 1 < n^3 \) holds true. This means we have to determine for what value of \( n \), the quadratic expression on the left hand side is less than the cubic expression on the right hand side.

Rearranging the Inequality

Let's subtract \( 2n^2 + 3n + 1 \) from both sides to bring all terms into one inequality: \( n^3 - 2n^2 - 3n - 1 > 0 \). We need to determine the smallest natural number \( n \) that satisfies this inequality.

Testing Small Values for n

We will test small values for \( n \), starting from \( n = 1 \), and check each one until the inequality is satisfied:- For \( n = 1 \): \( 1^3 - 2 \times 1^2 - 3 \times 1 - 1 = 1 - 2 - 3 - 1 = -5 \)- For \( n = 2 \): \( 2^3 - 2 \times 2^2 - 3 \times 2 - 1 = 8 - 8 - 6 - 1 = -7 \)- For \( n = 3 \): \( 3^3 - 2 \times 3^2 - 3 \times 3 - 1 = 27 - 18 - 9 - 1 = -1 \)- For \( n = 4 \): \( 4^3 - 2 \times 4^2 - 3 \times 4 - 1 = 64 - 32 - 12 - 1 = 19 \)

Conclusion on Smallest n

By testing values for \( n \), we found that for \( n = 4 \), the inequality \( n^3 - 2n^2 - 3n - 1 > 0 \) holds true as it results in a positive 19. None of the previous integers satisfied the inequality, meaning the smallest natural number \( n \) that satisfies the inequality is \( n = 4 \).

Key concepts

Understanding Natural Numbers

The set of natural numbers, denoted by \( \mathbb{N} \), includes all positive integers starting from 1: 1, 2, 3, and so on. They do not include zero or negative numbers. Natural numbers are the most basic type of numbers and are often used for counting and ordering.
In the exercise, we need to find the smallest natural number \( n \) that satisfies a given inequality. Since natural numbers are discrete and increase one by one, we test each one starting from the smallest (\( n = 1 \)) to find when the inequality becomes true. This approach is straightforward but can be tedious in more complex scenarios.
It is important to recognize that natural numbers serve as the fundamental building blocks in various mathematical disciplines, especially when dealing with algebraic and number-theoretic problems.

Exploring Cubic Equations

Cubic equations involve variables raised to the third power, like the expression \( n^3 \) in our inequality. A cubic equation generally has the form \( ax^3 + bx^2 + cx + d = 0 \) where \( a \), \( b \), \( c \), and \( d \) are constants and \( a eq 0 \).
Cubic equations can have one to three real roots or solutions, and their graphs create distinct curves. Understanding these curves can help in analyzing the behavior of the equation.In our exercise, the inequality \( n^3 - 2n^2 - 3n - 1 > 0 \) involves comparing a cubic expression against a quadratic one. Such inequalities are common in problem-solving. Their solutions often require testing values or using algebraic techniques to simplify and solve.Cubic equations can convey complex relationships but often reveal interesting solutions when broken down. They have applications that extend into engineering, physics, and higher-level math.

Insight into Quadratic Expressions

Quadratic expressions take the form of \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are coefficients, and \( a eq 0 \). These expressions represent a parabola on a graph, showing how the quadratic can open upwards or downwards.
The original inequality \( 2n^2 + 3n + 1 < n^3 \) includes a quadratic expression on the left-hand side, requiring us to find where it is less than the cubic expression on the right-hand side.Quadratics are commonly solved by finding their roots through factoring, using the quadratic formula, or completing the square. In inequalities involving quadratics, understanding the sign and value of the expression in its domain is essential.Recognition of how quadratics behave simplifies testing conditions in inequalities. These expressions are also prevalent in real-world scenarios, such as projectile motion and financial calculations.