Problem 28 - Precalculus with Limits

Question

In Exercises 23 - 32, find a formula for for the arithmetic sequence. \( 10, 5, 0, -5, -10, \cdots \)

Step-by-Step Solution

Answer

The formula for the given arithmetic sequence is \( a_n = 10 - 5 * (n-1) \)

Step 1: Identify the First Term

In the given arithmetic sequence, the first term \( a \) is 10.

Step 2: Identify the Common Difference

From observation, we see that the common difference \( d \) of the sequence is -5 as every term is reduced by 5 from the previous one.

Step 3: Formulate the Formula

Substituting the values of \( a \) and \( d \) into the formula, we get \( a_n = 10 + (n-1) * -5 \)

Key Concepts

Common Difference

An arithmetic sequence is defined by a constant difference between consecutive terms, known as the common difference. The common difference, denoted by \(d\), in an arithmetic sequence is fundamental to identifying how the sequence progresses. In the sequence given: 10, 5, 0, -5, -10, each term is decreasing by 5. This means that

• 5 - 10 = -5
• 0 - 5 = -5
• -5 - 0 = -5
• -10 - (-5) = -5

As you can see the common difference is -5. Understanding the common difference helps in finding any particular term in the sequence or even predicting what the sequence looks like farther down the line. The common difference can be positive, negative, or zero, altering the behavior and structure of the sequence.

Sequence Formula

To find a particular term in an arithmetic sequence, we use the sequence formula. The sequence formula for the \(n\)-th term, \(a_n\), in an arithmetic sequence is:\[a_n = a + (n-1) \cdot d\]This formula relies on the following:

• \(a\) - the first term of the sequence.
• \(d\) - the common difference between the terms.
• \(n\) - the position of the term in the sequence you are trying to find.

For the sequence 10, 5, 0, -5, -10, the first term \(a\) is 10 and the common difference \(d\) is -5. Plugging these values into the formula, you get:\[a_n = 10 + (n-1) \cdot (-5)\]This formula allows you to compute any term in the sequence efficiently. Just replace \(n\) with the term number you wish to find.

First Term

The first term of an arithmetic sequence, often denoted by \(a\), serves as the starting point for the sequence. In any arithmetic sequence, the first term is crucial because it, along with the common difference, sets the tone for the entire sequence. For the sequence presented in the exercise, the first term \(a\) is 10. This is the number from which all subsequent terms are calculated by repeatedly adding the common difference.For example, in an arithmetic sequence:

• If the first term is 3 and the common difference is 4, the sequence starts at 3, then becomes 7, 11, 15, and so forth.
• With a first term of 10 and a common difference of -5, as in our sequence, additions of -5 gradually decrease each term starting from 10.

Recognizing the first term is essential for correctly building or deconstructing the sequence to find any desired terms or properties.