Problem 72 - Algebra 2

Question

Write an explicit and a recursive formula for each arithmetic sequence. $$ 17,8,-1, \ldots $$

Step-by-Step Solution

Answer

The explicit formula for the arithmetic sequence is \( a_n = 17 + (n-1)(-9) \) and the recursive formula is \( a_n = a_{n-1} - 9 \).

Step 1: Identify the Common Difference

First, let's identify the common difference (d) in the given sequence. This can be done by subtracting the first term from the second term in the sequence, or the second term from the third, and so on. In this case, subtraction of the second term (8) from the first term (17), gives d=-9.

Step 2: Formulate the Explicit Formula

Let's formulate the explicit formula, which states the nth term as a direct function of n. Using the formula \( a_n = a_1 + (n-1)d \), and inserting the first term of the sequence (17) and the common difference (-9), we get \( a_n = 17 + (n-1)(-9) \).

Step 3: Formulate the Recursive Formula

Formulate the recursive formula. This formula expresses each term of the sequence as a function of its preceding term. For an arithmetic sequence, the recursive formula is \( a_n = a_{n-1} + d \). Using the given first term (17) and the common difference (-9), the recursive formula becomes \( a_n = a_{n-1} - 9 \).

Key Concepts

Explicit Formula

An explicit formula provides a direct way to find any term of an arithmetic sequence without having to list all the preceding terms. It is particularly useful because it allows you to jump straight to the desired term number, denoted usually as \(n\).

The standard form for the explicit formula of an arithmetic sequence is \( a_n = a_1 + (n-1)d \). Here, \(a_n\) is the nth term you're solving for, \(a_1\) is the first term in the sequence, and \(d\) is the common difference.

For example, in the sequence given: 17, 8, -1,...

• The first term, \(a_1\), is 17.
• The common difference, \(d\), is -9.

To form the explicit formula, substitute these values into the formula to find any term in the sequence:
\[ a_n = 17 + (n-1)(-9) \]

This formula tells us the value of the nth term is directly calculable, avoiding the need to work through each preceding term.

Recursive Formula

The recursive formula for an arithmetic sequence offers a way to find a term based on the one preceding it. It's like building a chain, where each link is dependent on the one before.

The recursive formula in general form is expressed as \( a_n = a_{n-1} + d \), where \(a_{n-1}\) is the previous term, and \(d\) is the common difference.

In the sequence 17, 8, -1,..., establish the following elements:

• The initial term, \(a_1\), starts you off with 17.
• The common difference, \(d\), is -9.

Thus, the recursive formula can be written as:
\[ a_n = a_{n-1} - 9 \]
This equation indicates how to compute each new term by subtracting 9 from the previous term.

Common Difference

At the heart of an arithmetic sequence lies the common difference, which is a constant value added to each term to get to the next. It essentially defines the step size between terms.

Finding the common difference (d) is simple. You subtract any term from the term that follows it. In our example sequence—17, 8, -1—the work is clear.

Begin with the subtraction:

• From 17 to 8 gives: \(8 - 17 = -9\)
• From 8 to -1 gives: \(-1 - 8 = -9\)

This consistent result confirms that the common difference \(d = -9\).
The common difference's value dictates how the sequence evolves. If positive, the sequence increases; if negative, as in our example, it decreases.