Problem 36 - Precalculus with Limits

Question

In Exercises 33-36, find the indicated term of the sequence. \( a_n = \dfrac{4n^2 - n + 3}{n(n - 1)(n + 2)} \) \( a_{13} = \Box \)

Step-by-Step Solution

Answer

The 13th term of the sequence, \( a_{13} \), is \( \dfrac{111}{390} \).

Step 1: Understand the formula

The sequence formula \( a_n = \dfrac{4n^2 - n + 3}{n(n - 1)(n + 2)} \) has \( n \) as the variable which denotes the position of each term in the sequence. Therefore, to find \( a_{13} \), replace \( n \) with 13 in the equation.

Step 2: Substitute the value of n

Substitute \( n = 13 \) into the sequence formula: \( a_{13} = \dfrac{4(13)^2 - 13 + 3}{13(13 - 1)(13 + 2)} \)

Step 3: Simplify the substituted formula

Carry out the operations in the numerator and the denominator separately: \( a_{13} = \dfrac{4(169) - 13 + 3}{13(12)(15)} = \dfrac{676 - 13 + 3}{2340} = \dfrac{666}{2340} \)

Step 4: Simplify the fraction to its lowest terms

The fraction \( \dfrac{666}{2340} \) can be simplified by dividing the numerator and the denominator by 6, which gives \( a_{13} = \dfrac{111}{390} \).

Key Concepts

Algebraic Simplification

Algebraic simplification is the process of making an algebraic expression easier to comprehend and solve, often by removing any redundancies or unnecessary components. In the exercise, algebraic simplification comes into play when dealing with the sequence formula. The formula given \[ a_n = \frac{4n^2 - n + 3}{n(n - 1)(n + 2)} \] requires substituting the given value of \( n \), which is 13, to find \( a_{13} \).
Simplification here begins by calculating each part of the expression separately, especially when involving powers and multiplication. Simplifying the expression often involves:

• Combining like terms in the numerator and denominator.
• Performing calculations such as multiplication and addition within the expression.
• Reducing the expression to a simpler form that is easier to interpret.

Simplification helps ensure accuracy and ease of calculation, making the process of identifying specific terms in sequences more efficient.

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. In the sequence formula\[ a_n = \frac{4n^2 - n + 3}{n(n - 1)(n + 2)} \], the numerator and the denominator themselves are polynomial expressions.
Rational expressions are crucial because they express relationships involving proportions and can be manipulated using various algebraic techniques. The key properties you can explore with rational expressions include:

• Substitution of numbers to evaluate the expression for specific values of variables.
• Detecting and excluding any undefined points (e.g., division by zero) in the expression.
• Using algebraic identities to simplify or factor the polynomials in the numerator or denominator.

Understanding how to work with rational expressions is essential for solving problems that involve polynomial divisions, especially in sequences and series like in this example.

Numerical Calculation

Numerical calculation involves performing arithmetic operations to evaluate expressions to reach a precise numerical outcome. After substituting \( n = 13 \) in the expression, the next step is to perform numerical calculations. This includes multiplying, adding, and subtracting numbers to simplify the equation to a solvable format.
In our specific solution, this entails:

• Substituting and calculating \( 4(169) \) in the numerator as \( 676 \).
• Computing other operations like subtraction and addition: \( 676 - 13 + 3 \).
• Calculating the denominator \( 13 \times 12 \times 15 \) which results in \( 2340 \).

Numerical calculations are essential as they translate algebraic expressions into concrete numbers that represent the desired term in a sequence.

Fraction Simplification

Fraction simplification, also known as reducing fractions, refers to the process of writing a fraction in its simplest form. To achieve this, factor both the numerator and the denominator, and eliminate all common factors. In the given exercise, you simplify the fraction \[ \frac{666}{2340} \]To do so:

• Identify the greatest common divisor (GCD) of the numerator and denominator, which is 6 in this case.
• Divide both the numerator and the denominator by their GCD.
• Simplify the fraction to \( \frac{111}{390} \).

Fraction simplification ensures that results are presented in their most understandable form, making computations and further operations much more manageable.