Problem 51 - College Algebra

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 \(a_n = \frac{1}{3}(n-1)\).

Step 1: Identify the Sequence Type

The sequence given is arithmetic, because each term increases by a constant difference. The arithmetic sequence starts with 0, followed by \(\frac{1}{3}\) and \(\frac{2}{3}\), indicating that the difference between consecutive terms is \(\frac{1}{3}\).

Step 2: Identify the Common Difference

The common difference \(d\) is the amount each term increases by. From the sequence \(0, \frac{1}{3}, \frac{2}{3}, \ldots\), we see that each term increases by \(\frac{1}{3}\). Therefore, \(d = \frac{1}{3}\).

Step 3: Identify the First Term

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

Step 4: Write the Explicit Formula

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

Step 5: Simplify the Formula

Simplifying the explicit formula, we get \(a_n = \frac{1}{3}(n-1)\). This formula can be used to find any term in the sequence.

Key Concepts

Explicit Formula

An explicit formula allows you to directly determine any term in an arithmetic sequence without needing to know the previous term. This type of formula is extremely handy because it gives you a direct calculation for any term position, \( n \).
The general form of the explicit formula for an arithmetic sequence is: \[ a_n = a_1 + (n-1) \cdot d \] Here,

• \( a_n \) represents the \( n \)-th term.
• \( a_1 \) signifies the first term of the sequence.
• \( n \) is the position of the term you want to find.
• \( d \) is the common difference between terms.

Using this explicit formula, you can compute the value of any term in the sequence by simply plugging in the appropriate values.

Common Difference

In an arithmetic sequence, the common difference is the constant value that separates successive terms. It essentially tells us how much each term in the sequence increases or decreases from the previous one.

To find the common difference \( d \):- Subtract any term in the sequence from the following term. - In the sequence given as \( 0, \frac{1}{3}, \frac{2}{3}, \ldots \), we find the common difference by calculating \( \frac{1}{3} - 0 = \frac{1}{3} \).

The consistency of the common difference is what defines an arithmetic sequence. It is always constant, ensuring the pattern remains linear as we move along the sequence.

First Term

The first term of an arithmetic sequence, denoted as \( a_1 \), is pivotal because it acts as the starting point for the entire sequence. It lays down the initial value from which the sequence grows or declines based on the common difference.

In the provided sequence, the first term is clearly \( 0 \).

• This value sets the baseline of the sequence.
• Using the explicit formula, each subsequent term is a build on this initial term.

Remember, identifying the first term accurately is crucial, as it is an integral part of the explicit formula that determines every other term in the sequence. Without knowing \( a_1 \), forming the sequence in its correct and original order becomes impossible.