Problem 51 - College Algebra with Corequisite Support

Question

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\left\\{0, \frac{1}{3}, \frac{2}{3}, \ldots\right\\} $$

Step-by-Step Solution

Answer

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

Step 1: Identify Initial Term

In an arithmetic sequence, the first term is denoted as \(a_1\). Here, the first term is \(0\). Hence, \(a_1 = 0\).

Step 2: Determine Common Difference

The common difference, \(d\), in an arithmetic sequence is the difference between any two consecutive terms. Calculate \(d = \frac{1}{3} - 0 = \frac{1}{3}\). Hence, the common difference is \(d = \frac{1}{3}\).

Step 3: Write the General Formula

The explicit formula for an arithmetic sequence is given by \(a_n = a_1 + (n-1) \cdot d\). Substitute the known values: \(a_n = 0 + (n-1) \cdot \frac{1}{3}\).

Step 4: Simplify the Explicit Formula

Simplify the expression from the previous step: \(a_n = (n-1) \cdot \frac{1}{3} = \frac{n-1}{3}\). Thus, the explicit formula for the sequence is \(a_n = \frac{n-1}{3}\).

Key Concepts

Understanding the Explicit Formula

An explicit formula in an arithmetic sequence is a powerful tool to find any term in the sequence without needing the previous one. The general form of an explicit formula for an arithmetic sequence is \(a_n = a_1 + (n-1) \cdot d\). This formula directly relates any term, \(a_n\), to:

• \(a_1\): the initial term
• \(n\): the term number you are finding
• \(d\): the common difference

By using the explicit formula, you can swiftly compute any term in the sequence. For instance, if you need the 10th term, you substitute 10 for \(n\) in the formula. This eliminates the need to manually calculate all preceding terms, streamlining the process efficiently.

Breaking Down the Common Difference

The common difference in an arithmetic sequence represents the consistent interval between consecutive terms. It is symbolized as \(d\) and calculated by subtracting a term from the next. As seen in the exercise, with the sequence \{0, \frac{1}{3}, \frac{2}{3}, \ldots\}, it's determined as:
\[d = \frac{1}{3} - 0 = \frac{1}{3}\]
This fixed increment is a defining feature of arithmetic sequences, ensuring that each term increases (or decreases) by this same measure. Understanding \(d\) helps you predict the behavior of the sequence and compute any term when combined with the initial term and term number.

• Adding \(d\) results in a sequence that grows.
• Subtracting \(d\) leads to a decreasing sequence.

Knowing \(d\) is crucial for mapping out the sequence neatly.

Defining the Initial Term

The initial term of an arithmetic sequence, designated as \(a_1\), serves as the starting point for the sequence. It is the first term, from which all subsequent terms are built by repeatedly adding the common difference. For the given problem, this is clearly identified as \(0\).
The role of \(a_1\) cannot be overstated as:

• It locates the beginning of the sequence.
• Directly influences the specific path or pattern of the sequence.
• Aids in the calculation of any term through the explicit formula.

The explicit formula's flexibility hinges on knowing the initial term, setting the foundation for locating any term with assured precision and ease.