Problem 31 - Precalculus with Limits

Question

In Exercises 23 - 32, find a formula for for the arithmetic sequence. \( a_3 = 94, a_6 = 85 \)

Step-by-Step Solution

Answer

The formula for the given arithmetic sequence is \(a_n = 100 + (n - 1)*(-3)\).

Step 1: Understanding the Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference of any two successive members is a constant. For an arithmetic sequence, the common difference \(d\) can be found by subtracting the previous term from the current term. It is represented as \(a_n = a_1 + (n - 1)*d\), where \(n\) is the number of the term, \(a_1\) is the first term and \(d\) is the common difference.

Step 2: Calculate the common difference

We have been given that \(a_3 = 94\) and \(a_6 = 85\). So, the difference between these two terms is \(a_6 - a_3 = 85 - 94 = -9\). However, as these are the 3rd and 6th terms, there are \(6 - 3 = 3\) terms between them. Therefore the common difference \(d\) is \( -9/3 = -3\).

Step 3: Find the first term

With the common difference and one term of the sequence, we can find the first term, \(a_1\). Using the term \(a_3 = 94\), we can plug the values into the general formula: \(a_n = a_1 + (n - 1)*d\), thus \(94 = a_1 + (3 - 1)*(-3)\). Solving for \(a_1\), we get \(a_1 = 100\).

Step 4: Build the formula for the arithmetic sequence

Now that we have the first term \(a_1 = 100\) and the common difference \(d = -3\), we can build the general formula of our sequence, which is \(a_n = 100 + (n - 1)*(-3)\).

Key Concepts

Common Difference

Understanding the concept of common difference is crucial when it comes to arithmetic sequences. In essence, the common difference is the consistent interval between consecutive terms in a sequence of numbers. This difference remains the same throughout the entire sequence and is denoted by the letter 'd'.

For instance, in the sequence 2, 5, 8, 11,... the common difference is 3, since each term increases by 3. If we were to consider a reverse scenario where numbers were decreasing, such as 10, 7, 4, ..., the common difference would be -3, indicating a decrease by 3 with each subsequent term. The formula to find the common difference is as simple as taking any term in the sequence (after the first) and subtracting the term that comes before it, that is,
\(d = a_{n} - a_{n-1}\).

Calculating Common Difference in Exercises

Students sometimes struggle with calculating the common difference when they aren't consecutive terms. In our exercise example where the 6th term \(a_6\) is 85 and the 3rd term \(a_3\) is 94, three terms separate them. To find the common difference, we take the difference between these terms and divide by the number of intervals between them, which results in \(d = \frac{85 - 94}{6 - 3} = -3\). The negative sign indicates that the sequence is decreasing.

Sequence of Numbers

A sequence of numbers is essentially an ordered list of numbers that follow a specific pattern or rule. The numbers in a sequence are often referred to as terms. Sequences can be finite or infinite, depending on whether they have a last term or continue indefinitely.

Arithmetic sequences, where each term is produced by adding a common difference to the previous term, are among the most straightforward types of sequences. Other types of sequences include geometric sequences, where each term is found by multiplying the previous term by a constant, or more complex sequences that may involve alternate operations or rules.

Identifying Patterns

The key to mastering sequences is recognizing the pattern that governs them. In an arithmetic sequence, once the pattern of addition or subtraction by the common difference is identified, any term in the sequence can be calculated. Using our exercise's arithmetic sequence, one might notice how each term decreases by -3, starting from \(a_3\) (94), to reach \(a_6\) (85), and this pattern would continue consistently throughout the terms of the sequence.

Arithmetic Progression

Arithmetic progression is another term for an arithmetic sequence. It is a series of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the previous term. Arithmetic progression is one of the simplest and most widely understood types of numerical patterns.

Arithmetic progression can be represented in two main ways: by listing out the terms, as in 3, 6, 9, 12, 15, ..., or by using a formula that expresses any term in the sequence as a function of its position, which is the standard form \(a_n = a_1 + (n - 1)d\), where \(a_n\) represents the nth term, \(a_1\) the first term, and 'n' the term's position in sequence.

Applying the Arithmetic Progression Formula

To solve problems involving arithmetic progression effectively, understanding how to use the formula is crucial. The exercise we examined demonstrates how to apply this understanding to determine the formula for a sequence when given non-consecutive terms. We used the given terms to find common difference and then the first term. Once we determined that the first term \(a_1 = 100\) and the common difference \(d = -3\), we were able to construct the general formula for the entire sequence: \(a_n = 100 + (n - 1)(-3)\). This formula now allows us to compute any term within the sequence.