Problem 88 - College Algebra

Question

Which one of the following is true? a. \(\frac{n !}{(n-1) !}=\frac{1}{n-1}\) b. The Fibonacei sequence \(1,1,2,3,5,8,13,21,34,55,89\) \(144, \ldots\) can be defined recursively using \(a_{0}=1, a_{1}=1\) \(a_{n}=a_{n-2}+a_{n-1},\) where \(n \geq 2\). c. \(\sum_{i=1}^{2}(-1)^{i} 2^{i}=0\) d. \(\sum_{i=1}^{2} a_{i} b_{i}=\sum_{i=1}^{2} a_{i} \sum_{i=1}^{2} b_{i}\)

Step-by-Step Solution

Answer

Out of the four given options, only option B is true.

Step 1: Evaluate Option A

Firstly evaluate \(\frac{n !}{(n-1)!}\). Factorial of any number \(n\) is defined as the product of all positive integers less than or equal to \(n\). By observing this, \(n!\) can be seen as \(n*(n-1)!\). Therefore, when we divide \(n!\) by \((n-1)!\), we get \(n\), not \(\frac{1}{n-1}\). Therefore, option A is false.

Step 2: Evaluate Option B

Fibonacci sequence is the series of numbers in which each number is the sum of the two preceding ones, usually starting with 0 and 1. The recursive formula given in option B, \(a_{n}=a_{n-2}+a_{n-1}\), fits perfectly with the definition of the Fibonacci sequence given above. Therefore, option B is true.

Step 3: Evaluate Option C

The summation given by \(\sum_{i=1}^{2}(-1)^{i} 2^{i}\) can be calculated manually as follows: when \(i = 1\), the term equals \(-1 * 2 = -2\), and when \(i = 2\), the term equals \(1 * 4 = 4\). Adding these terms together, we get \(-2 + 4 = 2\), not \(0\). So, option C is false.

Step 4: Evaluate Option D

The statement \(\sum_{i=1}^{2} a_{i} b_{i} = \sum_{i=1}^{2} a_{i} \sum_{i=1}^{2} b_{i}\) suggests that the sum of products is equal to the product of sums. In general, this is not true for any arbitrary sequences \(a\) and \(b\). Therefore, option D is also false.

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics often represented by the symbol \(!\). A factorial of a non-negative integer \(n\) is the product of all positive integers less than or equal to \(n\). This means that the factorial of \(n\), written as \(n!\), is calculated as:

• \(n! = n \times (n-1) \times (n-2) \times ... \times 1\)

When dealing with factorials, an interesting property arises. If you consider \(\frac{n!}{(n-1)!}\), this actually simplifies to \(n\). This is because in the fraction, every term from \(1\) to \(n-1\) cancels out, leaving just \(n\). This property makes calculations involving factorials much simpler, no wonder it is frequently used in combinations and permutations. Remember, factorials are only defined for non-negative integers, and \(0!\) is a special case that equals \(1\).

Fibonacci Sequence

The Fibonacci sequence is a famous series where each number is the sum of the two preceding ones, typically starting with every classical math student's favorite two numbers: 0 and 1. However, in some definitions, it can start with 1 and 1.

• The sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34...

An important aspect of the Fibonacci sequence is its recursive definition. This recursive formula allows you to calculate the sequence as follows:

• \(a_0 = 0, a_1 = 1\)
• \(a_n = a_{n-1} + a_{n-2}\)

This recursive relationship is precisely what defines the sequence. It means each term beyond the second is formed by adding the two terms preceding it. This mathematical beauty is not just limited to theoretical concepts but can also be observed in natural phenomena such as the branching of trees and arrangement of leaves.

Summation

Summation is a concise way of adding a sequence of numbers. The symbol \(\sum\) represents this operation, and it is defined over a specified index range. For example, \(\sum_{i=1}^{n} a_i\) means "sum the sequence \(a_i\) from \(i=1\) to \(i=n\)."

• An example computation is: \(\sum_{i=1}^{3} i = 1 + 2 + 3 = 6\).

In our exercise example, the summation \(\sum_{i=1}^{2}(-1)^{i} 2^{i}\) involves more complex terms:

• \(i = 1\): \((-1)^1 \times 2^1 = -2\)
• \(i = 2\): \((-1)^2 \times 2^2 = 4\)
• Total: \(-2 + 4 = 2\)

Understanding summations is key in algebraic reasoning, calculus, and various fields due to its streamline ability to sum rows of elements. Different bounds, sequences, and expressions can vastly change the resulting sum.

Recursive Formulas

Recursive formulas provide one of the most elegant ways to define sequences and structures in mathematics. They allow you to express each term in the sequence based on one or more preceding terms. It's like saying, "to find the next step, just look back a bit."

• The beauty of recursion is found in sequences like the Fibonacci sequence where you calculate each number as the sum of the two before it.
• In equations, they help solve problems by breaking them into smaller, connected pieces.

A recursive formula is generally written in the form:

• \( a_{n} = f(a_{n-1}, a_{n-2}, ..., a_{n-k}) \)

This tells us how to get \(a_n\) by applying function \(f\) to previous terms. Understanding recursive definitions are crucial for computer science and mathematical proofs, as they often provide a natural fit for iterative and looping constructs in programming. Their simplicity and power make recursion a favorite tool among mathematicians and engineers alike.