Problem 21 - Calculus

Question

Suppose \(f_{n}\) has the form \(a n+b n^{2}+c n^{3}\). If you know \(f_{1}=1, f_{2}=5, f_{3}=14,\) turn those into three equations for \(a, b, c .\) The solutions $a=\frac{1}{6}, b=\frac{1}{2}, c=\$$ give what formula?

Step-by-Step Solution

Answer

The formula is \( f(n) = \frac{1}{6} n + \frac{1}{2} n^2 + \frac{1}{3} n^3 \).

Step 1: Understanding the problem

We have been given a function in the form \( f(n) = a n + b n^2 + c n^3 \) and specific values for \( f_1, f_2, \) and \( f_3 \). We need to derive equations to find \( a, b, \) and \( c \) based on these values.

Step 2: Substituting to find first equation

Substitute \( n = 1 \) into the function to get the first equation: \[ f_1 = 1 = a imes 1 + b imes 1^2 + c imes 1^3 \] This simplifies to: \[ a + b + c = 1 \]

Step 3: Substituting to find second equation

Substitute \( n = 2 \) into the function to get the second equation: \[ f_2 = 5 = a imes 2 + b imes 2^2 + c imes 2^3 \] This simplifies to: \[ 2a + 4b + 8c = 5 \]

Step 4: Substituting to find third equation

Substitute \( n = 3 \) into the function to get the third equation: \[ f_3 = 14 = a imes 3 + b imes 3^2 + c imes 3^3 \] This simplifies to: \[ 3a + 9b + 27c = 14 \]

Step 5: Solving the system of equations

Now, we have three linear equations:1. \( a + b + c = 1 \)2. \( 2a + 4b + 8c = 5 \)3. \( 3a + 9b + 27c = 14 \)Solve this system of equations to find \( a, b, \) and \( c \). The solutions provided are \( a=\frac{1}{6}, b=\frac{1}{2}, c=\frac{1}{3} \).

Step 6: Formulating the final expression

Substitute the values of \( a, b, \) and \( c \) back into the expression for \( f(n) \): \[ f(n) = \frac{1}{6} n + \frac{1}{2} n^2 + \frac{1}{3} n^3 \]This is the formula for the function \( f(n) \).

Key Concepts

Linear Equations

Linear equations are the simplest form of equations in mathematics, and they represent relationships where the power of variables is always one. They often have a standard form of \( ax + b = 0 \), where \( a \) and \( b \) are constants. In our particular exercise, each equation we derived from substituting values into \( f(n) = a n + b n^2 + c n^3 \) has a more complex appearance because they are polynomial equations, not just linear. However, when considering the equations in terms of our unknowns \( a, b, \) and \( c \), they become a system of linear equations.

• First equation: \( a + b + c = 1 \)
• Second equation: \( 2a + 4b + 8c = 5 \)
• Third equation: \( 3a + 9b + 27c = 14 \)

Solving these helps us find the coefficients that define our polynomial function. Understanding linear equations is crucial because they serve as a foundation block for solving more complex mathematical problems.

Function Notation

Function notation is a way to express ideas in a concise and organized manner in mathematics. It generally uses notation like \( f(x) \), which stands for a function named \( f \) that is defined in terms of a variable \( x \). In our exercise, the function is described as \( f(n) = a n + b n^2 + c n^3 \), which specifies that \( f(n) \) is a function of \( n \). Why use function notation?

• It clarifies which variables are inputs and which symbols or letters stand for a functional relationship.
• It allows different functions to be treated independently, even in complex problems.
• It simplifies communication of mathematical relationships between variables.

By substituting specific values of \( n \) such as 1, 2, and 3 in \( f(n) \), we can derive different specific expressions or equations, showcasing the power of function notation to express different scenarios elegantly.

System of Equations

A system of equations is a set of equations with multiple variables. The task is to find values for the variables that satisfy all the equations simultaneously. In our exercise, we derived three distinct equations from the polynomial expression provided, forming a system:

• \( a + b + c = 1 \)
• \( 2a + 4b + 8c = 5 \)
• \( 3a + 9b + 27c = 14 \)

To solve this system of equations, we use methods such as substitution, elimination, or matrix operations to find the unknown coefficients \( a, b, \) and \( c \).

• Substitution: Solve one equation for one variable and substitute into the other equations.
• Elimination: Add or subtract equations to eliminate one variable, making it easier to solve for the others.
• Matrix Operations: Use techniques from linear algebra, such as row reduction, to solve systems more efficiently.

Mastering the concept of systems of equations allows students to tackle more complex mathematical problems that occur in various applications in science and engineering.