Problem 37 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 37-38: (a) Evaluate the given formula for the stated values of \(n\), and solve the resulting system of equations for \(a, b, c\), and \(d\). (This method can sometimes be used to obtain formulas for sums.) (b) Compare the result in part (a) with the indicated exercise, and explain why this method does not prove that the formula is true for every \(n\). $$ 1^{2}+2^{2}+3^{2}+\cdots+n^{2}=a n^{3}+b n^{2}+c n ; $$ \(n=1,2,3\) (Exercise 9)

Step-by-Step Solution

Answer

Determine \(a, b, c\) by solving equations for \(n = 1, 2, 3\); method limits verification to specific \(n\).

Step 1: Plug in Given Values for n

We are given the equation \(1^{2}+2^{2}+3^{2}+\cdots+n^{2}=a n^{3}+b n^{2}+c n\). We need to find the coefficients \(a, b, c\) by plugging in the given values: \(n=1, 2, 3\).

Step 2: Evaluate for n = 1

Substitute \(n = 1\) into the formula. Calculate the left side: \(1^2 = 1\). The equation becomes \(1 = a(1)^3 + b(1)^2 + c(1)\), or \(1 = a + b + c\).

Step 3: Evaluate for n = 2

Substitute \(n = 2\) into the formula. Calculate the left side: \(1^2 + 2^2 = 1 + 4 = 5\). The equation becomes \(5 = a(2)^3 + b(2)^2 + c(2)\), or \(5 = 8a + 4b + 2c\).

Step 4: Evaluate for n = 3

Substitute \(n = 3\) into the formula. Calculate the left side: \(1^2 + 2^2 + 3^2 = 1 + 4 + 9 = 14\). The equation becomes \(14 = a(3)^3 + b(3)^2 + c(3)\), or \(14 = 27a + 9b + 3c\).

Step 5: Solve the System of Equations

We now have the system: \(\begin{align*} 1 &= a + b + c, \ 5 &= 8a + 4b + 2c, \ 14 &= 27a + 9b + 3c. \end{align*}\). Solve this system to find \(a, b, c\). Use substitution or elimination to solve it.

Step 6: Verify and Compare

Once \(a, b, c\) are found, compare the solution to the result in Exercise 9, if available. This method evaluates certain instances and does not prove the entire formula for all \(n\). It shows these equations may hold for specific \(n\) but doesn't verify them universally.

Key Concepts

Polynomial Expressions

Polynomial expressions are mathematical expressions that involve variables raised to whole number powers and multiplied by coefficients. In essence, they are sums of several terms with different degrees. For instance, the expression given in the problem is a polynomial:

- It is written as: \(a n^{3} + b n^{2} + c n\)- Each term involves a variable \(n\), which is raised to different powers: \(n^3, n^2,\) and \(n^1\).- The coefficients, \(a, b, c\), are constants that multiply each respective term. Understanding polynomial expressions is crucial because they form the basis for many equations and theorems in mathematics, ranging from simple algebraic expressions to more complex calculus problems.

When working with polynomials like the one from the exercise, it's important to be familiar with the operations you can perform on them:

• Simplifying: Combine like terms (terms with the same power of \(n\)) to make the expression as concise as possible.
• Adding/Subtracting: Combine the coefficients of like terms when polynomials are added or subtracted.
• Multiplying: Use techniques such as distributive property or special formulas (like the square of a sum) to multiply polynomials.

Mastering these basics will provide a solid foundation to tackle more advanced topics like solving polynomial equations and mathematical proofs.

Solving Equations

Solving equations is a fundamental concept in mathematics, especially when it comes to systems of equations like in the given exercise.

In this context, a system of equations is a set of equations with multiple variables that you solve simultaneously. From the exercise, we have the system:- \(1 = a + b + c\)- \(5 = 8a + 4b + 2c\)- \(14 = 27a + 9b + 3c\)
To solve such systems, we often use methods like substitution or elimination:

• Substitution: Solve one of the equations for one variable and plug that expression into the other equations. Continue this until you isolate one variable.
• Elimination: Add or subtract equations to eliminate one variable, making it easier to solve for the remaining variables.

These methods help find the values of \(a, b,\) and \(c\) that make all equations in the system true simultaneously.

After solving, verifying your results by substituting them back into the original equations ensures that they are correct. This verification step helps in confirming that the derived solution satisfies all equations involved, maintaining the integrity of your solution process.

Mathematical Proof

Mathematical proof is a logical argument that establishes the truth of a statement beyond doubt. In the context of the exercise, while evaluating the equation for specific values of \(n\) provides evidence for these cases, it does not constitute a complete proof for all possible values of \(n\).

In mathematics, to prove a statement universally, methods such as:

• Induction: Prove the base case, then assume it holds for a particular \(n\), and show it must also hold for \(n+1\).
• Direct Proof: Use axioms and previously proven statements to directly show the truth of a statement.
• Contradiction: Assume the statement is false and show that this assumption leads to a logical contradiction.

These techniques are crucial in mathematics because they validate whether a result is true in a general sense, not limited to specific instances or trials.

The exercise highlights that evaluating a formula for particular numbers can suggest its validity, but without a formal proof technique, we cannot assume it for all integers \(n\). Thus, it's essential to understand the difference between testing specific cases and proving universally—all mathematical truths require rigorous proof to stand.