Problem 70 - College Algebra

Question

Calculate the first eight terms of the sequences \(a_{n}=\frac{(n+2) !}{(n-1) !}\) and \(b_{n}=n^{3}+3 n^{2}+2 n,\) and then make a conjecture about the relationship between these two sequences.

Step-by-Step Solution

Answer

The two sequences are identical for the first eight terms.

Step 1: Understanding the Formula for Sequence 1

The first sequence is defined by the formula \(a_n = \frac{(n+2)!}{(n-1)!}\). This formula uses factorials, where \(n!\) represents the product of all positive integers up to \(n\). Our task is to calculate the first eight terms of this sequence.

Step 2: Calculating Terms for Sequence 1

For each term, \(n = 1\) to \(8\):- For \(n=1\), \(a_1 = \frac{(1+2)!}{(1-1)!} = \frac{3!}{0!} = \frac{6}{1} = 6\).- For \(n=2\), \(a_2 = \frac{(2+2)!}{(2-1)!} = \frac{4!}{1!} = \frac{24}{1} = 24\).- Similarly, calculate the terms: - \( a_3 = \frac{5!}{2!} = 60 \) - \( a_4 = \frac{6!}{3!} = 120 \) - \( a_5 = \frac{7!}{4!} = 210 \) - \( a_6 = \frac{8!}{5!} = 336 \) - \( a_7 = \frac{9!}{6!} = 504 \) - \( a_8 = \frac{10!}{7!} = 720 \).

Step 3: Understanding the Formula for Sequence 2

The second sequence is defined by the formula \(b_n = n^3 + 3n^2 + 2n\). This formula involves straightforward polynomial operations. Our task is to calculate the first eight terms of this sequence.

Step 4: Calculating Terms for Sequence 2

For each term, \(n = 1\) to \(8\):- For \(n=1\), \(b_1 = 1^3 + 3 \times 1^2 + 2 \times 1 = 1 + 3 + 2 = 6\).- For \(n=2\), \(b_2 = 2^3 + 3 \times 2^2 + 2 \times 2 = 8 + 12 + 4 = 24\).- Similarly, calculate the terms: - \( b_3 = 3^3 + 3 \times 3^2 + 2 \times 3 = 60 \) - \( b_4 = 4^3 + 3 \times 4^2 + 2 \times 4 = 120 \) - \( b_5 = 5^3 + 3 \times 5^2 + 2 \times 5 = 210 \) - \( b_6 = 6^3 + 3 \times 6^2 + 2 \times 6 = 336 \) - \( b_7 = 7^3 + 3 \times 7^2 + 2 \times 7 = 504 \) - \( b_8 = 8^3 + 3 \times 8^2 + 2 \times 8 = 720 \).

Step 5: Observing the Patterns

Compare each term from both sequences: \(a_1 = b_1 = 6\), \(a_2 = b_2 = 24\), ..., \(a_8 = b_8 = 720\). Observe that each corresponding term is exactly equal in both sequences.

Step 6: Making a Conjecture

Based on the calculations, the conjecture is that the two sequences \(a_n\) and \(b_n\) produce identical values for each term.

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics used to simplify complex expressions and solve various problems. A factorial, denoted as \( n! \), is the product of all positive integers up to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials grow quickly as \( n \) increases, making them significant in combinatorics and probability.
In the sequence formula \( a_n = \frac{(n + 2)!}{(n - 1)!} \), factorials allow us to compute large products conveniently. The term \( (n + 2)! \) encompasses more multiplications than \( (n - 1)! \), which makes this sequence interesting as it simplifies to a manageable expression for each \( n \). Understanding how factorials work will help you break down similar mathematical problems with efficiency.

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. The general form is \( a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \).
The formula for the second sequence, \( b_n = n^3 + 3n^2 + 2n \), is a polynomial of degree 3. With polynomials, each term represents a piece of the overall solution influenced by \( n \) raised to different powers. Understanding polynomials helps unravel various mathematical problems by identifying patterns and predicting behaviors of sequences over an extended range of values.

Combinatorial Identities

Combinatorial identities often involve factorials and binomial coefficients, and they are crucial in counting problems. They establish equivalences between different counting expressions.
While our problem did not explicitly use combinatorial identities, understanding how to manipulate factorials like in \( a_n = \frac{(n+2)!}{(n-1)!} \) can relate to these identities. These manipulations represent larger sets of combinations and permutations, fundamental to theoretical and applied combinatorics. Recognizing equivalent forms, like those found in identities, aids in reducing the complexity of many problems.

Conjecture

A conjecture is an educated guess or hypothesis about a mathematical relationship or pattern. It is based on observations and initial evidence but requires formal proof to be established as a theorem.
In analyzing the sequences \( a_n \) and \( b_n \), we observed that their terms were identical for all \( n \) up to 8. This leads to the conjecture that the two sequences produce the same outputs indefinitely. Making such conjectures is an essential part of mathematical exploration and problem-solving. It prompts deeper investigation and proof attempts to verify or refute the observed relationship.

Patterns in Sequences

Patterns in sequences help identify regularities or repeated elements within a given series of numbers. Recognizing patterns is critical for predicting subsequent elements and understanding underlying structures.
In this problem, both sequences \( a_n \) and \( b_n \) exhibited identical terms, revealing a symmetric pattern. Identifying such patterns can simplify problems, allow for more generalized solutions, and foster conjectures about mathematical behavior. In practice, observing these patterns often leads to discoveries of deeper principles and new mathematical relationships.