Problem 26 - Precalculus with Limits

Question

In Exercises 23 - 32, find a formula for for the arithmetic sequence. \( a_1 = 0, d = -\dfrac{2}{3} \)

Step-by-Step Solution

Answer

The formula for the given arithmetic sequence is \( a_n = \dfrac{-2}{3} \cdot (n-1) \)

Step 1: Identify Given Values

Here, the first term (denoted by \( a_1 \)) of the arithmetic sequence is given as 0, and the common difference (denoted by \( d \)) between the terms is \(-\dfrac{2}{3}\). These are all the values we need to write the general formula for an arithmetic sequence.

Step 2: Write Down the Formula for an Arithmetic Sequence

The general formula for the \( n^{th} \) term (\( a_n \)) of an arithmetic sequence is given by: \( a_n = a_1 + (n-1) \cdot d \). Here, \( n \) is the term number, \( a_1 \) is the first term, and \( d \) is the common difference.

Step 3: Substitute the Given Values into the Formula

By substituting the given values into the formula, we get: \( a_n = 0 + (n-1) \cdot -\dfrac{2}{3} = \dfrac{-2}{3} \cdot (n-1) \)

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is an essential component that defines how each term is related to its preceding term. It is denoted by the letter "d" and represents the consistent difference you add to the previous term to obtain the current term.
For instance, if you have an arithmetic sequence, and each term is 2 more than the previous one, then the common difference is 2.

• It determines the sequence's pattern.
• Can be a positive number, making the sequence increase, or a negative number, causing it to decrease.
• Helps to calculate any term in the sequence when combined with the first term.

In the exercise provided, the common difference is \(-\frac{2}{3}\). This means each term is \(-\frac{2}{3}\) less than the term before it. Understanding this value helps in formulating the general sequence.

General Term Formula

The general term formula is a powerful tool in arithmetic sequences. It allows you to find any term in the sequence without listing all the previous ones. The formula is given by:\[ a_n = a_1 + (n-1) \, d \]where:

• \(a_n\) is the term you want to find.
• \(a_1\) is the first term in the sequence.
• \(d\) is the common difference.
• \(n\) is the position of the term in the sequence.

By using this formula, you can jump straight to the term number you're interested in, without calculating each previous term.
For example, in the exercise, the first term \(a_1\) is 0, and using a common difference of \(-\frac{2}{3}\), we can find any \(n^{th}\) term by plugging these values into the formula.

Sequence Formula

The sequence formula is crucial for understanding and generating the terms of an arithmetic sequence based on known values. It provides a structured way to continue, find, and interpret the sequence. The sequence of terms in an arithmetic sequence, derived from the general term formula \[ a_n = a_1 + (n-1) \, d \], starts with the first term and adds the common difference repeatedly.
For the sequence starting with \(a_1 = 0\) and having a common difference of \(-\frac{2}{3}\), the formula becomes:\[ a_n = 0 + (n-1) \left(-\frac{2}{3}\right) \]This transforms into:\[ a_n = -\frac{2}{3}(n-1) \]Using this formula, we can construct the entire sequence: \{0, -\frac{2}{3}, -\frac{4}{3}, -2, \ldots\}. Each term builds off the previous one by subtracting \(-\frac{2}{3}\). This formula solidifies our understanding of how each element is related.