Problem 35 - College Algebra

Question

For the following exercises, write a recursive formula for each arithmetic sequence. $$ a=\left\\{\frac{1}{5}, \frac{9}{20}, \frac{7}{10}, \ldots\right\\} $$

Step-by-Step Solution

Answer

The recursive formula is \(a_n = a_{n-1} + \frac{1}{4}\) with initial condition \(a_1 = \frac{1}{5}\).

Step 1: Identify the Common Difference

First, observe the given arithmetic sequence \(a = \left\{ \frac{1}{5}, \frac{9}{20}, \frac{7}{10}, \ldots \right\}\). Calculate the common difference \(d\) by subtracting the first term from the second term. Convert \(\frac{1}{5}\) to \(\frac{4}{20}\) to have a common denominator for easier subtraction: \(\frac{9}{20} - \frac{4}{20} = \frac{5}{20} = \frac{1}{4}\). So, \(d = \frac{1}{4}\).

Step 2: Write the First Term of the Sequence

The first term of the sequence \(a_1\) is the first number in the sequence, which is \(a_1 = \frac{1}{5}\).

Step 3: Formulate the Recursive Formula

A recursive formula for an arithmetic sequence is generally given as \(a_n = a_{n-1} + d\). For this sequence, substitute \(d = \frac{1}{4}\). Thus, the recursive formula is \(a_n = a_{n-1} + \frac{1}{4}\).

Step 4: Specify the Initial Condition

To fully define the recursive sequence, specify the initial condition: \(a_1 = \frac{1}{5}\). This allows for calculating further terms in the sequence starting from \(a_1\).

Key Concepts

Arithmetic Sequences

Arithmetic sequences are a series of numbers where the difference between consecutive terms is constant. This means that you can predict the next number in the sequence by adding this constant value, known as the common difference, to the previous term. This predictable pattern makes arithmetic sequences a key concept in mathematics, especially when dealing with series and patterns. For instance, in the sequence \( a = \left\{ \frac{1}{5}, \frac{9}{20}, \frac{7}{10}, \ldots \right\} \), each term after the first is obtained by consistently adding the same amount. This pattern helps us to construct a rule that describes the entire sequence.

Common Difference

The common difference in an arithmetic sequence is the fixed amount added to each term to arrive at the next term. To find this difference, you subtract any term from the term that follows it. For example, in the sequence \( a = \left\{ \frac{1}{5}, \frac{9}{20}, \frac{7}{10}, \ldots \right\} \), convert \( \frac{1}{5} \) to \( \frac{4}{20} \) to make subtraction easier. Then, calculate \( \frac{9}{20} - \frac{4}{20} = \frac{5}{20} = \frac{1}{4} \). As a result, the common difference \( d \) is \( \frac{1}{4} \). Knowing the common difference is crucial for understanding the sequence's behavior and forming its recursive formula.

Initial Condition

The initial condition is the starting point of the sequence, and it is essential for generating the sequence using a recursive formula. It provides the first term from which all following terms are calculated. In our sequence example \( a = \left\{ \frac{1}{5}, \frac{9}{20}, \frac{7}{10}, \ldots \right\} \), the initial condition is \( a_1 = \frac{1}{5} \). Setting this initial condition ensures that when using the recursive formula, you always have a base term to build from. Without the initial condition, you can't determine any subsequent terms, making it impossible to fully describe the sequence.

Sequence Terms

Sequence terms are the individual elements that make up a sequence. In an arithmetic sequence, each term can be defined using its position in the sequence and the recursive formula. For example, using the recursive formula \( a_n = a_{n-1} + \frac{1}{4} \), each term depends on the one that precedes it. You start with the initial condition, which in this case is \( a_1 = \frac{1}{5} \). Calculating further terms involves simply applying the common difference repeatedly:

• \( a_2 = \frac{1}{5} + \frac{1}{4} = \frac{9}{20} \)
• \( a_3 = \frac{9}{20} + \frac{1}{4} = \frac{7}{10} \)

These individual terms showcase the impact of the initial condition and the common difference.