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Chapter 11: Problem 70

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.

Short Answer

Expert verified
The sequences \(a_n\) and \(b_n\) are equal for the first eight terms, suggesting \(a_n = b_n\).

Step by step solution

01

Understanding the Problem

We need to calculate the first eight terms for each of the sequences given: \(a_n = \frac{(n+2)!}{(n-1)!}\) and \(b_n = n^3 + 3n^2 + 2n\). We'll compute the terms for \(n = 1\) to \(n = 8\).
02

Calculate Terms for Sequence \(a_n\)

To find each term \(a_n\), substitute \(n\) from 1 to 8 into \(a_n = \frac{(n+2)!}{(n-1)!}\). Simplify the expression by canceling out the factorials where possible:- \(a_1 = \frac{3!}{0!} = 6\)- \(a_2 = \frac{4!}{1!} = 24\)- \(a_3 = \frac{5!}{2!} = 60\)- \(a_4 = \frac{6!}{3!} = 120\)- \(a_5 = \frac{7!}{4!} = 210\)- \(a_6 = \frac{8!}{5!} = 336\)- \(a_7 = \frac{9!}{6!} = 504\)- \(a_8 = \frac{10!}{7!} = 720\).
03

Calculate Terms for Sequence \(b_n\)

Calculate each term \(b_n\) by substituting \(n\) from 1 to 8 into \(b_n = n^3 + 3n^2 + 2n\):- \(b_1 = 1^3 + 3 \cdot 1^2 + 2 \cdot 1 = 6\)- \(b_2 = 2^3 + 3 \cdot 2^2 + 2 \cdot 2 = 24\)- \(b_3 = 3^3 + 3 \cdot 3^2 + 2 \cdot 3 = 60\)- \(b_4 = 4^3 + 3 \cdot 4^2 + 2 \cdot 4 = 120\)- \(b_5 = 5^3 + 3 \cdot 5^2 + 2 \cdot 5 = 210\)- \(b_6 = 6^3 + 3 \cdot 6^2 + 2 \cdot 6 = 336\)- \(b_7 = 7^3 + 3 \cdot 7^2 + 2 \cdot 7 = 504\)- \(b_8 = 8^3 + 3 \cdot 8^2 + 2 \cdot 8 = 720\).
04

Compare the Sequences' Terms

After calculating the terms, compare them to look for patterns. Notice that for each \(n\) from 1 to 8, the terms of \(a_n\) and \(b_n\) are equal:- Both sequences have the terms 6, 24, 60, 120, 210, 336, 504, 720 respectively for \(n = 1\) to 8.
05

Make a Conjecture

Based on our calculations, we can conjecture that the terms of the sequences \(a_n\) and \(b_n\) are equal for all natural numbers \(n\), thus \(a_n = b_n\). This suggests that \(\frac{(n+2)!}{(n-1)!} = n^3 + 3n^2 + 2n\) holds for these values of \(n\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factorials
A factorial is a product of all positive integers from 1 up to a given number. Noted as "n!", the factorial multiplies a series of descending natural numbers. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Factorials grow very quickly with increasing \(n\).

Key features of factorials:
  • \(0!\) is defined as 1, which is essential for simplifying expressions.
  • The factorial of a number is often used in permutations and combinations, statistics, and various math fields.
  • The growth of factorials can make them computationally intensive for large \(n\).
In our exercise, the sequence \(a_n\) involves factorials in the form \((n+2)!/(n-1)!\). This expression reduces nicely because of the cancelation that happens due to the factorial property \((n+2)\times(n+1)\times(n)!\). This allows us to simplify calculations efficiently, especially for small values of \(n\).
Cubic Polynomials
Cubic polynomials are algebraic expressions involving a variable raised to the third power. The general form of a cubic polynomial in one variable, \(x\), is \(ax^3 + bx^2 + cx + d\), where \(a eq 0\).

Characteristics of cubic polynomials:
  • Cubic polynomials can have up to three real roots (or solutions).
  • They describe a curve that can involve one or two bends depending on the values of their coefficients.
  • The fundamental theorem of algebra guarantees a cubic polynomial has exactly three roots in the complex number field.
In the polynomial sequence \(b_n=n^3 + 3n^2 + 2n\), we replaced each \(n\) with the numbers 1 through 8, resulting in sequences that interestingly match the terms derived from the factorial-based sequence \(a_n\). This highlights a deep relationship between different forms of mathematical expressions.
Conjecture
A conjecture is an educated guess or a hypothesis made based on observed patterns. In mathematics, conjectures often arise from identifying consistent results or trends before being rigorously proved through logical deduction.

Features of a conjecture include:
  • It stems from empirical evidence but lacks formal proof.
  • Some famous conjectures, like the Goldbach Conjecture, have driven extensive research.
  • Conjectures remain significant for eventual formal proof or counterexample.
In our case, by observing the sequences \(a_n\) and \(b_n\) yielding identical terms for \(n = 1\) to 8, we formulated the conjecture that the sequences are equal for all natural numbers \(n\). This often suggests deeper mathematical truths linking seemingly different mathematical structures, such as factorials and polynomials.

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