Problem 63 - College Algebra with Corequisite Support

Question

List the first six terms of the sequence \(a_{n}=\frac{n^{3}-3.5 n^{2}+4.1 n-1.5}{2.4 n}\).

Step-by-Step Solution

Answer

The first six terms are approximately 0.0417, 0.1458, 0.875, 2.385, 4.708, and 7.854.

Step 1: Understand the Term Formula

The sequence is given by \( a_{n} = \frac{n^{3} - 3.5n^{2} + 4.1n - 1.5}{2.4n} \). To find the terms, substitute \( n \) with the integers starting from 1.

Step 2: Calculate the First Term

Substitute \( n = 1 \) into the formula: \( a_{1} = \frac{1^{3} - 3.5 \times 1^{2} + 4.1 \times 1 - 1.5}{2.4 \times 1} = \frac{1 - 3.5 + 4.1 - 1.5}{2.4} = \frac{0.1}{2.4} \approx 0.0417 \).

Step 3: Calculate the Second Term

Substitute \( n = 2 \) into the formula: \( a_{2} = \frac{2^{3} - 3.5 \times 2^{2} + 4.1 \times 2 - 1.5}{2.4 \times 2} = \frac{8 - 14 + 8.2 - 1.5}{4.8} = \frac{0.7}{4.8} \approx 0.1458 \).

Step 4: Calculate the Third Term

Substitute \( n = 3 \) into the formula: \( a_{3} = \frac{3^{3} - 3.5 \times 3^{2} + 4.1 \times 3 - 1.5}{2.4 \times 3} = \frac{27 - 31.5 + 12.3 - 1.5}{7.2} = \frac{6.3}{7.2} \approx 0.875 \).

Step 5: Calculate the Fourth Term

Substitute \( n = 4 \) into the formula: \( a_{4} = \frac{4^{3} - 3.5 \times 4^{2} + 4.1 \times 4 - 1.5}{2.4 \times 4} = \frac{64 - 56 + 16.4 - 1.5}{9.6} = \frac{22.9}{9.6} \approx 2.385 \).

Step 6: Calculate the Fifth Term

Substitute \( n = 5 \) into the formula: \( a_{5} = \frac{5^{3} - 3.5 \times 5^{2} + 4.1 \times 5 - 1.5}{2.4 \times 5} = \frac{125 - 87.5 + 20.5 - 1.5}{12} = \frac{56.5}{12} \approx 4.708 \).

Step 7: Calculate the Sixth Term

Substitute \( n = 6 \) into the formula: \( a_{6} = \frac{6^{3} - 3.5 \times 6^{2} + 4.1 \times 6 - 1.5}{2.4 \times 6} = \frac{216 - 126 + 24.6 - 1.5}{14.4} = \frac{113.1}{14.4} \approx 7.854 \).

Key Concepts

Algebraic Formulas

Algebraic formulas play a crucial role in sequences, serving as structured expressions that define a sequence's rule. For this particular sequence, the formula is given as \( a_{n} = \frac{n^{3} - 3.5n^{2} + 4.1n - 1.5}{2.4n} \).
These formulas often consist of polynomial expressions in the numerator. In this case, the numerator is a cubic polynomial \( n^{3} - 3.5n^{2} + 4.1n - 1.5 \), while the denominator is a linear term \( 2.4n \).

Such formulas allow you to calculate any term of the sequence once you have the index \( n \). This power of algebraic formulas lies in their ability to consistently and precisely define each term based on its position, ensuring predictable and repeatable calculations across different terms in the sequence.

Term Calculation

Term calculation involves substituting values into an established formula to find specific terms. Here, you would substitute \( n \) with integers starting from 1 to find terms like \( a_1, a_2, ..., a_6 \).
Let's delve into how this substitution works using our sequence as an example:

• For \( n = 1 \), calculate \( a_1 = \frac{1^{3} - 3.5 \times 1^{2} + 4.1 \times 1 - 1.5}{2.4 \times 1} \).
• Continue substituting successive integer values for \( n = 2, 3, ... , 6 \).

As these substitutions occur, the calculation simplifies each expression comprehensively to provide numerical terms of the sequence. Remember always to resolve each operation in the correct order, which is usually parentheses, exponents, multiplication/division (from left to right), and finally addition/subtraction (the PEMDAS order).
Accurate term calculation is critical in sequences to ensure each term aligns with the pattern dictated by the algebraic formula.

Polynomial Sequences

Polynomial sequences involve sequences defined by polynomials, where each term is derived from substituting sequence positions into the polynomial. Our sequence has a numerator polynomial, \( n^{3} - 3.5n^{2} + 4.1n - 1.5 \), which is a third-degree polynomial due to the highest power of the cube.

Characteristics of polynomial sequences:

• They can exhibit varying rates of increase or decrease based on the polynomial's degree and coefficients.
• The behavior and shape of polynomial sequences heavily rely on the polynomial's highest degree. Higher degrees often result in steeper increases or decreases as \( n \) grows.

In understanding the sequence's growth pattern, paying attention to the polynomial's leading term and its sign helps predict how the sequence behaves as it progresses.
For polynomial sequences, comprehending how each polynomial component contributes to the overall sequence will allow for deeper insights into the sequence's nature and future terms.