Problem 37 - College Algebra with Corequisite Support

Question

For the following exercises, write a recursive formula for each arithmetic sequence. $$ a=\left\\{\frac{1}{6},-\frac{11}{12},-2, \ldots\right\\} $$

Step-by-Step Solution

Answer

Recursive formula: \( a_{n} = a_{n-1} - \frac{13}{12} \) with \( a_1 = \frac{1}{6} \).

Step 1: Identify the First Term

The first term of the arithmetic sequence, denoted as \( a_1 \), is given directly in the sequence. From the series \( \left\{ \frac{1}{6}, -\frac{11}{12}, -2, \ldots \right\} \), we identify \( a_1 = \frac{1}{6} \).

Step 2: Calculate the Common Difference

The common difference \( d \) of an arithmetic sequence is found by subtracting the first term from the second term. Use the formula \( d = a_2 - a_1 \), where \( a_2 = -\frac{11}{12} \). Thus, \( d = -\frac{11}{12} - \frac{1}{6} \). First, convert \( \frac{1}{6} \) to \( \frac{2}{12} \) to have a common denominator: \( d = -\frac{11}{12} - \frac{2}{12} = -\frac{13}{12} \).

Step 3: Write the Recursive Formula

The recursive formula for an arithmetic sequence is given by \( a_{n} = a_{n-1} + d \), where \( n \geq 2 \) and \( d \) is the common difference. Using the first term and the common difference we found, the recursive formula becomes \( a_{n} = a_{n-1} - \frac{13}{12} \) with \( a_1 = \frac{1}{6} \).

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a countable collection of numbers that exhibit a pattern characterized by the addition of a constant value, known as the common difference, to arrive at the next term. This pattern or sequence is quite linear in nature, making it easy to predict subsequent numbers once the initial term and the common difference are determined. Key characteristics of an arithmetic sequence include:

• Each term is generated by adding or subtracting the same fixed number to the previous term.
• If you know a specific number in the sequence and the common difference, you can calculate any other number in the sequence.
• The arithmetic sequences can be increasing, decreasing, or constant, depending on the common difference.

Arithmetic sequences are used in various mathematical and real-world scenarios, such as financial calculations and analyzing patterns.

Common Difference

The common difference in an arithmetic sequence is the key element that maintains the uniformity in the progression of the numbers. It is a consistent value that gets added (or subtracted if negative) as you keep moving to the next term in the sequence. To find the common difference:

• Subtract the first term from the second term: Use formula \( d = a_2 - a_1 \).
• Ensure that fractions are simplified correctly. For example, converting \( \frac{1}{6} \) to \( \frac{2}{12} \) allows you to work easily with a common denominator.

A negative common difference, as in the example \( -\frac{13}{12} \), signifies the sequence is decreasing. In contrast, a positive difference indicates an increasing sequence.

First Term

The first term of an arithmetic sequence sets the initial point from which the sequence builds its subsequent numbers. It is often denoted by \( a_1 \) and is presented as the very first number in the sequence. Knowing the first term is crucial because it serves as the base value for calculating the rest of the sequence when combined with the common difference.In mathematical problems:

• The first term can be uniquely identified in a given sequence of numbers. For instance, in the sequence \( \left\{ \frac{1}{6}, -\frac{11}{12}, -2, \ldots \right\} \), the first term is \( a_1 = \frac{1}{6} \).
• It forms part of the recursive formula used to determine other terms in the sequence.

Recognizing and correctly identifying the first term is vital for resolving sequence-related questions, especially those involving calculations of future terms.

Sequence Formulas

Sequence formulas are mathematical expressions used to define sequences, enabling the prediction and calculation of any term within a sequence. Specifically, an arithmetic sequence uses specific types of sequence formulas, such as recursive or explicit formulas. The recursive formula defines a sequence where each term is determined by one or more of the preceding terms in the sequence. For an arithmetic sequence, the recursive formula looks like:

• \( a_n = a_{n-1} + d \) for all \( n \geq 2 \)
• \( a_1 \) is given as the first term

This approach is helpful for step-by-step construction of the sequence, making it easy to spot trends or errors.While the recursive formula is dynamic and efficient for sequences and programming, it must be supported by knowing the initial term, \( a_1 \), and the common difference, \( d \). Thus, having a clear understanding of the formula assists in sequence analysis and problem-solving in diverse applications.