Problem 36 - College Algebra

Question

For the following exercises, write a recursive formula for each arithmetic sequence. $$ a=\left\\{-\frac{1}{2},-\frac{5}{4},-2, \ldots\right\\} $$

Step-by-Step Solution

Answer

The recursive formula is \( a_n = a_{n-1} - \frac{3}{4} \).

Step 1: Identify the First Term

The first term in the sequence is given as \( a_1 = -\frac{1}{2} \). This is where we start our recursive formula.

Step 2: Calculate the Common Difference

To find the common difference \( d \) of an arithmetic sequence, subtract the first term from the second term: \( d = -\frac{5}{4} - \left(-\frac{1}{2}\right) = -\frac{5}{4} + \frac{2}{4} = -\frac{3}{4} \).

Step 3: Write the Recursive Formula

In a recursive formula for an arithmetic sequence, each term \( a_n \) is expressed as the sum of the previous term \( a_{n-1} \) and the common difference \( d \). Therefore, our recursive formula becomes: \( a_n = a_{n-1} - \frac{3}{4} \).

Key Concepts

Recursive Formula

A recursive formula is a method to describe the elements of a sequence based on preceding terms. In arithmetic sequences, the relationship is described by expressing each term as a function of its preceding term. For an arithmetic sequence, this formula takes the form: \[ a_n = a_{n-1} + d \] Here, \( a_n \) represents the current term, \( a_{n-1} \) is the preceding term, and \( d \) is the common difference.

• It allows us to generate any term if we know the previous term and the common difference.
• This type of formula is especially useful for sequences with a clear repetitive pattern.

Do note that the recursive formula requires an initial term to start the sequence.

Common Difference

The common difference is a fundamental aspect of arithmetic sequences. This is the fixed amount that separates one term in the sequence from the next. Calculating it involves subtraction:\[ d = a_2 - a_1 \]Here, \( a_2 \) is the second term, and \( a_1 \) is the first term. For our arithmetic sequence example, the common difference \( d \) is calculated as:\[ d = -\frac{5}{4} - \left(-\frac{1}{2}\right) = -\frac{3}{4}\]

• It can be positive or negative, impacting whether the sequence is increasing or decreasing.
• This value is consistent throughout the sequence.

By knowing the common difference, it becomes straightforward to predict subsequent terms based on the known pattern.

First Term

The first term of an arithmetic sequence, often denoted as \( a_1 \), is crucial because it sets the starting point of the sequence. In this context, the first term for our sequence is:\[ a_1 = -\frac{1}{2} \]

• Every sequence needs an initial point to properly function, providing reference for the subsequent terms.
• This initial term must be known to apply the recursive formula.

Understanding the starting point helps the sequence unfold correctly according to the recursive formula.