Question

Show for \(n=0,1,2\) that $$(n+1)(2 n+1)(2 n+3) / 3+(2 n+3)^{2}=(n+2)(2 n+3)(2 n+5) / 3$$

Short answer

The given identity holds true for the values of \( n = 0, 1, \) and \( 2 \).

Step-by-step solution

Plug in n = 0 into the equation

Substitute \( n = 0 \) into the equation: \[ (0+1)(2(0)+1)(2(0)+3)/3 + (2(0)+3)^2 = (0+2)(2(0)+3)(2(0)+5)/3 \]Simplifying, this becomes:\[ (1)(1)(3)/3 + 3^2 = (2)(3)(5)/3 \]\[ 1 + 9 = 10 \]\[ 10 = 10 \]Thus, the equation holds true for \( n = 0 \).

Plug in n = 1 into the equation

Substitute \( n = 1 \) into the equation:\[ (1+1)(2(1)+1)(2(1)+3)/3 + (2(1)+3)^2 = (1+2)(2(1)+3)(2(1)+5)/3 \]Simplifying, this becomes:\[ (2)(3)(5)/3 + 5^2 = (3)(5)(7)/3 \]\[ 10 + 25 = 35 \]\[ 35 = 35 \]Thus, the equation holds true for \( n = 1 \).

Plug in n = 2 into the equation

Substitute \( n = 2 \) into the equation:\[ (2+1)(2(2)+1)(2(2)+3)/3 + (2(2)+3)^2 = (2+2)(2(2)+3)(2(2)+5)/3 \] Simplifying, this becomes:\[ (3)(5)(7)/3 + 7^2 = (4)(7)(9)/3 \]\[ 35 + 49 = 84 \]\[ 84 = 84 \]Thus, the equation holds true for \( n = 2 \).

Key concepts

Algebraic Equations

Algebraic equations form the backbone of many mathematical problems and their solutions. An algebraic equation is essentially a statement that two expressions are equal. In the given exercise, we have a complex algebraic equation involving terms with factors and powers of \( n \).

The exercise is set up to verify the equality of the expressions on both sides of the equation for several values of \( n \). This specific task involves verifying, via substitution, whether the equation holds true. Algebraic equations like these appear frequently in various fields of mathematics, such as calculus and algebra, because they allow us to systematically solve for unknowns by ensuring the left-hand side equals the right-hand side.

Understanding and solving algebraic equations require grasp over basic mathematical operations like addition, multiplication, and exponentiation. Additionally, being familiar with the properties of numbers, such as distributive, associative, and commutative properties, can aid in the simplification of these equations for easier handling and verification.

Mathematical Proofs

Mathematical proofs are a fundamental part of mathematics and are used to demonstrate that a statement or proposition is always true. In this exercise, the task is to show that the given equation holds for specific values of \( n \). This type of problem often employs what is known as a **proof by exhaustion**, where each potential case is checked individually.

Proofs by exhaustion work by confirming the validity of a statement for each possibility, leaving no room for error as all scenarios are covered. This approach is particularly effective when the number of cases is manageable, such as in this example where we only need to check three values \( n = 0, 1, \) and \( 2 \).

The **purpose of proofs** is to ensure the reliability of mathematical claims. Without proofs, mathematics would lack rigor and reliability. Proving statements helps us in making mathematical reasoning more robust and ensures that claimed patterns or equalities are not just coincidental but have a firm foundation.

Substitution Method

The substitution method is a technique used in solving equations and proving their consistency. In the context of this exercise, the substitution method involves replacing the variable \( n \) with specific values to simplify and verify the original algebraic equation.

Through substitution, the task entails slotting in \( n = 0, 1, \) and \( 2 \) into our equation. After substituting these numbers, the next step is simplifying the expressions on both sides and ensuring they are equal, thus confirming the correctness of the equation for those values. This step-by-step substitution allows for a simple and effective way to check the truth of complex statements without solving the entire problem initially.

The substitution method serves as a powerful tool in checking identities, working through polynomial equations, and solving systems of equations. Moreover, it provides a clear path from a complicated expression to straightforward arithmetic calculations, making it a favored approach in mathematical simplification exercises.