Problem 13 - Contemporary Precalculus

Question

Find the first five terms of the sequence \(\left\\{a_{n}\right\\}\). \(a_{n}\) is the \(n\) th digit in the decimal expansion of \(\pi\)

Step-by-Step Solution

Answer

Answer: The first five terms of the sequence are 1, 4, 1, 5, and 9.

Step 1: Recall the decimal expansion of \(\pi\)

The first few decimal places of \(\pi\) are 3.14159...

Step 2: Extract the decimal digits as the elements of the sequence

The first five decimal digits after the decimal point are 1, 4, 1, 5, and 9. These will be the first five terms in the sequence \(\left\\{a_{n}\right\\}\).

Step 3: Write down the sequence

The first five terms of the sequence \(\left\\{a_{n}\right\\}\) are: 1. \(a_1 = 1\) 2. \(a_2 = 4\) 3. \(a_3 = 1\) 4. \(a_4 = 5\) 5. \(a_5 = 9\) Thus, the first five terms of the sequence are 1, 4, 1, 5, and 9.

Key Concepts

Decimal Expansion of Pi

Understanding the decimal expansion of pi is essential for many areas of mathematics and science. Pi, represented as \(\pi\), is an irrational number, which means it cannot be expressed as a simple fraction. Its decimal representation is infinite and never repeating, starting with 3.14159 and continuing without end.

The sequence of numbers that follow the decimal point (1, 4, 1, 5, 9, ...) represents the decimal expansion of \(\pi\). These individual digits can be considered as elements in a sequence. Each number in this sequence is known as a decimal place of \(\pi\), and each decimal place represents a term in the sequence of \(\pi's\) expansion.

When dealing with the decimal expansion of \(\pi\), it’s important to note that while we may use a limited number of decimal places for practical calculations, the true value of \(\pi\) is non-terminating. This non-repeating nature is part of what makes \(\pi\) so fascinating to mathematicians and students alike.

Mathematical Sequences

A mathematical sequence is a list of numbers arranged in a specific order, where each number is known as a 'term'. Sequences can either be finite, with a specific number of terms, or infinite, continuing indefinitely. The position of each term within a sequence is usually represented by an index number, denoted by \(n\), often starting with \(n = 1\).

Sequences are a fundamental concept in mathematics because they can describe patterns and can be defined by a formula, a method for finding the terms, or even recursively, in terms of previous terms in the sequence. For example, the decimal expansion of \(\pi\) can be seen as an infinite sequence, where each digit after the decimal point represents a term in the sequence. The terms of this expansion are strictly determined by the value of \(\pi\) itself, and each digit has a set place within the sequence, which is non-repeating and has no discernible pattern.

Defining Sequence Terms

To fully grasp the concept of defining sequence terms, one should understand how each term is formulated or identified within a sequence. For instance, in a sequence defined by a particular property or rule, such as the decimal expansion of \(\pi\), the \(n\)th term is determined by its position in the sequence.

Each term \(a_n\) in a sequence can be defined explicitly by a formula that describes the relationship between the term's value and its position \(n\). Take the sequence example given in the exercise, where the \(n\)th term \(a_n\) corresponds to the \(n\)th digit after the decimal point in \(\pi's\) decimal expansion. To define each term:

• The first term \(a_1\) is the first digit after the decimal point, which is 1.
• The second term \(a_2\) is the second digit after the decimal point, which is 4.
• This pattern continues for subsequent terms, such as \(a_3\) being the third digit, and so on.

The ability to define sequence terms is crucial because it allows for the identification of specific elements within a sequence and facilitates the understanding of the sequence's nature and behavior.