Problem 45 - Precalculus with Limits

Question

In Exercises 41 - 46, write the first five terms of the arithmetic sequence defined recursively. \( a_1 = \dfrac{5}{8}, a_{n + 1} = a_n - \dfrac{1}{8} \)

Step-by-Step Solution

Answer

The first five terms of the given arithmetic sequence are \(\frac{5}{8}, \frac{1}{2}, \frac{3}{8}, \frac{1}{4}, \frac{1}{8}\)

Step 1: Write Down the Initial Term

The first term, \(a_1\), is already provided: \(a_1 = \frac{5}{8}\). So it will be the starting point.

Step 2: Calculate the Second Term

Utilize the recursive formula \(a_{n + 1} = a_n - \frac{1}{8}\) to find the succeeding term. Substituting \(n = 1\) in the formula we get, \(a_2 = a_1 - \frac{1}{8} = \frac{5}{8} - \frac{1}{8} = \frac{4}{8} = \frac{1}{2}\)

Step 3: Calculate the Third Term

Again using the recursive formula, replace \(n = 2\). \(a_3 = a_2 - \frac{1}{8} = \frac{1}{2} - \frac{1}{8} = \frac{4}{8} - \frac{1}{8} = \frac{3}{8}\)

Step 4: Calculate the Fourth and Fifth Terms

Repeat the same process for \(n = 3\) and \(n = 4\), to get the fourth and fifth terms respectively. \(a_4 = a_3 - \frac{1}{8} = \frac{3}{8} - \frac{1}{8} = \frac{2}{8} = \frac{1}{4}\) and \(a_5 = a_4 - \frac{1}{8} = \frac{1}{4} - \frac{1}{8} = \frac{2}{8} - \frac{1}{8} = \frac{1}{8}\)

Key Concepts

Recursive Formula

A recursive formula is a mathematical expression that allows us to find the next term in a sequence by using the preceding term or terms. In other words, it's like having a set of instructions to follow to get from one term to the next. This is particularly useful in arithmetic sequences, where each term is generated by adding (or subtracting) a fixed number from the previous term.

In the exercise above, the recursive formula is given as:

• \[ a_{n + 1} = a_n - \frac{1}{8} \]

Here, this tells us that to find any term in our sequence, we simply need to subtract \( \frac{1}{8} \) from the term before it.

This recursive relationship creates a systematic way to determine all terms of the sequence, starting with the initial term.

Sequence Terms

Sequence terms are the individual elements or numbers in a sequence that follow a specific rule or pattern. In an arithmetic sequence, the terms are numbers that increase or decrease by a constant amount, known as the common difference.

To make sense of this concept, let's consider our sequence from the exercise:

• The first term, \( a_1 = \frac{5}{8} \)
• The second term, \( a_2 = \frac{1}{2} \)
• The third term, \( a_3 = \frac{3}{8} \)
• The fourth term, \( a_4 = \frac{1}{4} \)
• The fifth term, \( a_5 = \frac{1}{8} \)

Each term is derived directly from the term before it by applying the recursive formula outlined earlier. The step-by-step calculations show this logical progression, confirming each term in the sequence builds upon the last.

Common Difference

The common difference in an arithmetic sequence is the fixed amount you either add or subtract to get from one term to the next. It's like a stepping stone, keeping the sequence consistently moving in one direction.

In our exercise, the common difference is explicitly stated as \(-\frac{1}{8}\). This means each term is \(\frac{1}{8}\) less than the previous term, creating a sequence that decreases steadily.

The common difference:

• Determines how the sequence behaves over time.
• A positive common difference would mean the terms increase.
• A negative common difference (as in our example) leads the terms to decrease.

Understanding the common difference is crucial because it ensures we maintain the right pattern throughout the sequence, allowing us to predict future terms easily.