Problem 22 - Algebra and Trigonometry

Question

the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. $$ a_{n}=-2(n-1) ! $$

Step-by-Step Solution

Answer

The first four terms of the sequence defined by the given general term \(a_n = -2(n - 1)!\) are -2, -2, -4, -12 respectively.

Step 1: Understanding the Task

The sequence is defined by the general term \(a_n = -2(n-1)!\). We need to find the first four terms of the sequence. This translates into finding the values of the sequence for \(n = 1, 2, 3, 4\). Note that (n-1)! denotes the factorial of (n-1).

Step 2: Finding the First Term of the Sequence

First, let's find the first term of the sequence by substituting \(n = 1\) into the general term. This will give us \(a_1 = -2(1 - 1)! = -2(0)! = -2*1 = -2\), since 0! is defined to equal 1.

Step 3: Finding the Second Term of the Sequence

Next, let's find the second term by substituting \(n = 2\) into the general term. This yields \(a_2 = -2(2 - 1)! = -2(1)! = -2*1 = -2\), since 1! equals 1.

Step 4: Finding the Third Term of the Sequence

Then, substituting \(n = 3\) into the general term gives \(a_3 = -2(3 - 1)! = -2(2)! = -2*2 = -4\), since 2! is equal to \(2*1 = 2\).

Step 5: Finding the Fourth Term of the Sequence

Finally, substitution of \(n = 4\) into the general term gives \(a_4 = -2(4 - 1)! = -2(3)! = -2*6 = -12\), since 3! equals \(3*2*1 = 6\).

Key Concepts

Factorial Notation

Factorial notation is integral to understanding various concepts in mathematics, particularly in combinatorics and sequences. A factorial, represented using the exclamation point ( factorial of any non-negative integer, denoted as \(n!\), is the product of all positive integers less than or equal to \(n\). The factorial of zero, \(0!\), is a special case defined to be 1. Factorials grow very rapidly with increasing \(n\), which is a key characteristic to keep in mind when working with such sequences.

For instance, \(3! = 3 \times 2 \times 1 = 6\), and \(4! = 4 \times 3 \times 2 \times 1 = 24\). This concept is used in the provided exercise to evaluate the sequence's terms. It's also essential to remember that factorial notation only applies to whole numbers; it isn't defined for decimals or negative integers. Knowing how to work with factorials is crucial for the sequence in the exercise, as it forms the backbone of the algebraic expressions used to determine the terms.

Sequence of Numbers

In mathematics, a sequence is an ordered list of numbers that follow a particular rule or pattern. Each number in the sequence is referred to as a term. For the sequence given in the exercise, the general term is given by an algebraic expression involving a factorial, which determines the rule for creating the sequence.

Typically, sequences are defined by a formula for the nth term, such as \(a_n = -2(n-1)!\) in our case. The first four terms are found by substituting consecutive integer values for \(n\), starting with 1, into this formula. Sequences can be finite or infinite; they may exhibit patterns, such as arithmetic or geometric progressions, or they might be more complex, involving recursive definitions or factorial operations, as shown in the exercise. Intuitively understanding how to generate terms from the provided general term is an invaluable skill in exploring the properties and patterns within sequences.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations that represent a particular mathematical quantity. They do not contain equal signs, unlike equations. The algebraic expressions become especially interesting when they encompass factorials or other advanced mathematical operations as seen in the sequence exercise.

In the context of our exercise, the algebraic expression \(-2(n-1)!\) determines the value of each term in the sequence. It combines the arithmetic operations of multiplication and subtraction with the more complex operation of factorial. Being able to interpret and calculate values from such expressions is a fundamental aspect of algebra and helps students unravel patterns in sequences, solve for unknowns, and understand how different elements relate to one another within a mathematical framework.

Why Factorial Matters in Algebraic Expressions

Factorial notation has a profound impact on the value of algebraic expressions, particularly when sequences are involved. It amplifies the rate at which terms increase or decrease and introduces a level of complexity often encountered in permutations and combinations, series, and more advanced topics in mathematics. Grasping this relationship between factorials and algebraic expressions can aid in a deeper comprehension of the sequence involved and improve problem-solving techniques.