Question

tell whether the sequence is arithmetic. Explain your reasoning. \(81,27,9,3,1, \ldots\)

Short answer

No, the sequence \(81,27,9,3,1, \ldots\) is not arithmetic because the differences between consecutive terms are not constant.

Step-by-step solution

Understand the Arithmetic Sequence

An arithmetic sequence is one in which the difference between any two consecutive terms is constant. Arithmetic sequences progress by adding or subtracting the same value each time. This means that if we subtract the second element in the sequence from the first, the third from the second, the forth from the third and so on, we should get the same result each time if the sequence is arithmetic.

Check the Differences

Start by identifying the differences in the given sequence \(81,27,9,3,1, \ldots\) . For example, \(27 - 81 = -54\), \(9 - 27 = -18\), \(3 - 9 = -6\) and \(1 - 3 = -2\).

Analyze the Differences

Observe the differences between successive terms. If the differences are not constant and the people keep changing, then the sequence is not arithmetic. Here, the differences vary from -54, -18, -6, and -2. Therefore, the given sequence \(81,27,9,3,1, \ldots\) is not an arithmetic sequence.

Key concepts

Mathematical Sequences

In mathematics, a sequence is a list of numbers arranged in a specific order that often follows a unique pattern or rule. Sequences can be finite or infinite and are fundamental in understanding various mathematical concepts. They serve as the foundation for series and provide a basis for understanding functions and more complex mathematical constructs.

Some common types of sequences include arithmetic, geometric, harmonic, and Fibonacci sequences. Each of these sequences is defined by a specific rule that describes the relationship between its terms. For example, in an arithmetic sequence, the difference between any two consecutive terms is constant, which is known as the 'common difference'. In a geometric sequence, on the other hand, each term is found by multiplying the previous term by a constant value, known as the 'common ratio'.

Understanding the properties of sequences allows us to predict subsequent terms and write equations that describe the entire sequence. By mastering sequence concepts, we can tackle a wide array of problems in mathematics and apply these skills to various disciplines, including computer science, economics, and physics.

Constant Difference

The concept of 'constant difference' is a cornerstone of arithmetic sequences. It refers to the situation where the difference between each consecutive pair of terms in the sequence remains the same. This constant difference can be positive or negative, depending on whether the sequence is increasing or decreasing, respectively.

To illustrate this with numbers, consider the arithmetic sequence 3, 7, 11, 15, ..., where the constant difference is 4, because each term increases by 4 from the previous term. To identify if a given sequence is arithmetic, one must examine the differences between consecutive terms, as shown in the exercise solution steps. If these differences change from one pair of terms to another, the sequence is not arithmetic.

Importance of a Constant Difference

Having a constant difference simplifies calculations and predictions within the sequence. It allows us to construct a general formula for the nth term of an arithmetic sequence, typically expressed as \( a_n = a_1 + (n - 1)d \), where \( a_n \) is the nth term, \( a_1 \) is the first term, \( n \) is the position of the term in the sequence, and \( d \) is the constant difference. This formula is a powerful tool in algebraic reasoning and solving real-world problems involving sequential and patterned events.

Algebraic Reasoning

Algebraic reasoning is a logical thought process that involves identifying patterns, creating equations, and using symbolic representations to solve problems. This form of reasoning enables us to describe relationships between quantities and to generalize patterns using algebraic expressions and formulas.

In the context of sequences, algebraic reasoning helps us to define sequences with formulas, calculate specific terms, and even find the sum of all the terms in sequences. For instance, when working with a potential arithmetic sequence, algebraic reasoning is applied to verify the equality of differences between terms. If the differences are not equal, as in the exercise provided, algebraic reasoning allows us to conclude that the sequence is not arithmetic.

Application in Sequences

Algebraic reasoning extends beyond simply identifying types of sequences. It also plays a crucial role when we encounter more complex sequences or when sequences have practical applications, such as in financial calculations, computer algorithms, or modeling natural phenomena. With solid algebraic reasoning skills, learners can approach and solve a broad spectrum of problems efficiently and effectively, fostering mathematical understanding and competence that extends far beyond the classroom.